Let me preface by saying:

If you are new to Heimer, it is likely you will require repeated cry hurt to learn the way of things, and it is likely, that nothing I say will stop you from precluding that the champion is weak, and nonviable. You will be mistaken, but your wrongness will not be atypical, and THAT is what pains me, greatly.

Arranging the items like I did results in weird stat calculations above.

Actual stats amount to:
39% cooldown reduction at 18 as well as 572 AP with Deathcap Calculation but before ROA addition

Enjoy.

I have played Heimer through his various iterations. Here is a very solid way to go about it currently.

meki + 2 pots : start
fiendish + t1 boots + mana pot (if needed) : first back
fiendish + chalice + t1 boots : 2nd back
Athene's + t1 boots + Needlessly : 3rd back
Athene's + t1 boots + NLR + catalyst : 4th back
Athene's + t1 boots + NLR + catalyst + ward(s) : 5th back
Athene's + t1 boots + NLR + catalyst + Blasting Rod + MOAR WARD(s) : 6th back
Athene's + Sorcerer's Shoes + Deathcap + Catalyst + MOAR WARD(s) : 7th back
Athene's + SS + Deathcap + Rod of Ages + MOAR WARD(s) : 8th Back
Athene's + SS + Deathcap + ROA + Giant's Belt + MOAR WARD(s) : 9th Back
Athene's + SS + Deathcap + ROA + Rylai's + Void Staff + Consumeable Pots and Oracle : 10th

There are a couple flexible parts to this.
If the enemy team is stacking magic resist heavily, you might be better off getting Void Staff before Rylai's ... this all depends on how reliant your team is on your damage (and how much magic resist the enemy team is stacking), and how much crowd control your team really has when a fight breaks. You could probably substitute Zhonyas and drop either Rylais or Void Rod. It all depends how good you are at positioning, how much MR they are stacking and how much health buffer you find yourself needing to not get exploded if you make a mistake.

Depending on how the game is going and how quickly you can amass funds, you may or may not have to go back this many times. You may have to go back slightly more often should your team be giving an extended demonstration on how to fail completely.

I build Athene's first because:

1. Heimerdinger's cooldowns are long.
2. Heimerdinger is a mana hog.
3. Heimerdinger MUST have damage too!
4. Mana buffer only, will fail you when you need it most.

I've tried a variety of builds, and the standard build I see around basically asserts that you ought to focus on either huge damage, or health buffer, or both. Heimer's Ult Passive is NOT going to give you enough cooldown reduction to give you a pass on building for cooldown. You'll end up in those situations early -> Mid game where you're waiting on a cooldown to kill the **** with a rocket worth of health as he finishes you off or skitters happily back to base. I'm not going to lie, that is probably going to happen to you no matter what you do.. BUT STILL! Build for cooldowns first, mana, and damage by order of priority in the very beginning.

YOU WILL NOT HAVE ENOUGH DAMAGE OR UTILITY WITH ABILITY POWER AND HEALTH ONLY! GET SOME COOLDOWN ON THERE FAST... It's going to take til level 16 before your Cooldown Passive kick's in fully.

First rule of Heimer is:

You must know what every champion is capable of, or you're sunk. Plain and simple.

Beginning Game:
2 faerie charms + 1 pot : start

I take turret first every time. I use it sparingly as a defensive measure. Generally after leashing I start by trying to last hit as many minions as possible with auto attack, my primary concern is farming up to kage's pick as quickly as possible. Know what your lane opponent wants to happen, and avoid it. If they move toward you they are confident (be afraid), if they move away they are baiting you (be afraid), if they are doing what you are doing and farming passively or taking occasional potshots, they are either very smart, or very dumb. (be afraid) In the very beginning of the game, it's best to only use your turrets to farm if you find that your opponent is trying to push you back. If they start killing your turret, throw another one down before they last hit it. (they've wasted their time) If you can't get another one down, auto attack and wait it out. Let your turrets act as a deterrent, they aren't for farming early game, and they aren't for offense, they are a purely defensive weapon. Unless your jungler is very good, it's super unlikely you will either drive your lane opponent out of lane, or kill them early game. They would have to be very haughty and impale themselves on your turret(s) for you to force them back so soon. Early game you have two goals. Make every last hit you can, DO NOT DIE. If you're pushing past the middle of the lane in the beginning, you're seriously doing it wrong.

A little later:
fiendish + t1 boots + mana pot (if needed) : first back
fiendish + chalice + t1 boots + amplifying tome : 2nd back

If you've attained fiendish + chalice + t1 boots + amplifying tome without dying. You are in really good shape, and getting to mid game (also known as pwnville) should be a breeze. Mana shouldn't be much of a problem at this point. Most of the time when my jungler asks if I want blue, I advise him/her to take it instead. You will have some cooldown, a bunch of mana regen and some decent poking ability.

At this point you're going to step up your aggression against your lane opponent. Now you'll start using your double turrets to farm in mid. I position mine with 1 on either side of the lane at what would be the center of the river if it flowed through mid. If your opponent tries to focus down your turrets, use your ult to save them. If they take it out... put another one right down. They can't stop you from farming to your heart's content. Try and assist your turrets with last hits by wrenching minions at the last second to ensure they don't get killed by creeps. When your ult is off you shoot 3 rockets. Keep your eyes open for any time your opponent is one of the 3 closest hostiles to your current position. As soon as they are, ROCKETS. If you can put them in that position without unnecessarily opening yourself up to a river gank, ROCKETS. If they try and come toward you or their jungler comes in from river... Run halfway to your tower, put down a turret and hit your ult. As long as you didn't put yourself in a stupid position, and you're not too low on health, they will probably impale themselves on you. ROCKETS ROCKETS ROCKETS all the time ****ing ROCKETS!

Some of you may be wondering to yourself why I don't take grenades until level 10? I'll tell you why. They use a ****load of mana, it's a super unreliable stun, and an extra level in rockets is gauranteed punishment for your lane opponent. Furthermore, half the time you're facing someone in mid who is playing just as cautiously as you are.. and by now they have had a taste of your poking.... If you want them close enough to the middle of the lane to conveniently set up a gank for your jungler... you're probably going to have to be out of range of your grenade... And flashing in to MAYBE get a grenade stun is more likely to get you killed than assist in a kill. Your opponent is going to naturally start spacing themselves outside of rocket range pretty early.... You'll have to give them space to get them to make mistakes. Remember your rockets benefit from your ult, your grenade does not! It's also unlikely that you'll be doing much roaming in the bush before level 10 without your mates. IT IS TRUE, there have been times that I wished I had a grenade before level 10, I will admit it. I believe it is worth not getting. Grenades are fantastic for tossing at team fight blobs, and stunning people up close when they're chasing you through bushes or something. It takes a highly practiced Heimer to catch people with grenade on the run consistently, and an even more practiced Heimer to flash in from distance, and hit a fleeing target at nearly max gren range.... take it from me. You can make your own decision on the matter, but I suggest you don't knock it until you try it.

Beginning of Mid Game:
Athene's + t1 boots + Needlessly: 3rd back
Athene's + t1 boots + NLR + catalyst: 4th back
Athene's + t1 boots + NLR + catalyst + ward(s) : 5th back

Ok look, if you've made it this far without being killed, (And they've probably tried several times) you're doing well. You now have probably somewhere around 34% cooldown reduction, no mana problems, and a very respectable amount of damage. The one thing you don't have until your hypothetical 4th back, is health buffer. Once you get your giant's belt and a few wards, it's bloody feeding time.... but until then you are a glass ****ing cannon on the back of go cart.

TEAMFIGHT STRATEGY INTERMISSION:
Small engagement:
If favorable, throw grenade + rockets, run in and drop turrets + ult.... rinse repeat until enemies are dead or the tide turns and you have to run.

If unfavorable, throw grenade + rockets, start moving away drop turrets + ult.....
Wait to see how fatalistic the enemy team behaves and how many of them are left to chase, and at what health. If they start charging into your turrets to chase you or your friends like derpalerps... try and land a grenade on their face when they are in the midst of your turret fire, while on the run. KEEP YOUR DISTANCE!

One trick I like to use is if there's only 1 chasing, and there's a bush ahead. Run into the bush, drop a turret (or two if there's time) and wait for them. Hit them with a grenade when they get pretty much into or just in front of the bush, and rocket as well. This nets me so many kills it's absurd... even against bruisers and tanks. Landing the grenade is essential though, and knowing what's following behind them (and how far) is very important as well. You aren't trying to trade kills, you're aiming for victory and maximum humiliation. Concentrate! Only the center of the grenade stuns, and you need the stun for this! Good players may bait the bush by running close then abruptly turning around. You'll have to use your best judgement about whether they're tricking you or not. Some bushes simply aren't likely to be warded at certain times.

The decision whether or not to stand your ground and fight it out with someone, or run like a small wounded animal, is completely based on experience with enemy champs and what they can do... That's really what it comes down to. In almost every situation, if you don't 1. Get stun and 2. get a turret down in a 1v1 battle to the death, you are probably going to be the one doing the dying. Your turrets are sustained damage, Rocket + grenade is rarely enough to finish someone off up close and personal. It's always best to be camping 1 previously planted turret, before they get up in your grill. Then stun, then rocket, then second turret. Then they cry.

Ideally when a team fight breaks out, you'll lay down your turrets near or right along the edge of the fight (preferably covering your mates' exit route), and spam rockets and grenades into the fray until the entire enemy team is destroyed. If this happens it will probably result in total or partial victory.

If it does not happen, you'll be running, they'll be chasing. When someone is chasing after you, landing a grenade directly on their head is generally not the difference between life and death (if you ran early enough), but it's definitely the difference between locking them in your turret fire and scoring an easy kill, or watching them get away with shreds of health left when they realize their mistake. If I'm on the run, and I happen to score a decent grenade stun, often times I'll turn, rocket and plant another turret in range to slow their escape and go toe to toe (depending on what we both have for health and what they've got for crowd control). Grenade is easily one of if not the hardest stuns in the game to land, becoming good with it needs to be a priority, it is vital in mid - late game. Again, the low cooldowns in this build afford you a lot of running and gunning ability, once the fight has broken out, it's safe to say that everything you have should be on cooldown at all times or you're probably doing it wrong....

Bottom line, your job in the team fight is to explode the enemy team with rockets, blind/stun/damage by tossing grenades into groups of them (or individuals) and frustrate them by dividing / zoning the battlefield.

End Phase:
Athene's + Sorcerer's Shoes + Deathcap + Catalyst + MOAR WARD(s) : 7th back
Athene's + SS + Deathcap + Rod of Ages + MOAR WARD(s) : 8th Back
Athene's + SS + Deathcap + ROA + Giant's Belt + MOAR WARD(s) : 9th Back
Athene's + SS + Deathcap + ROA + Rylai's + Void Staff + Consumeable Pots and Oracle : 10th

During this time you'll probably have to make trips back to base, but won't have enough to buy the next component. Grab a couple wards when this is the case. They pay for themselves.

Once you get death cap you should either be well on your way toward smashing their doors in, or valiantly defending your base from invaders.

At death cap, you will likely be hitting them VERY hard. After Void Staff it gets pretty bad for them, particularly if they've been stacking magic resist up until now. After void things start to fall off a little bit though. Without a doubt their tanks will get tankier, and be much more of a threat to you getting pinned down. Again your job is to be on the outskirts of a team flight blob, trying to keep it under control, and pummeling with rockets and grenades. Keep an eye out for any opportunity you have for assisting teamates in escape with grenades and a convenient turret drop. ALWAYS take advantage of enemy teams that mass up on a single target by grenading them. Your single grenade will hit like a heavy tibbers, and your rocket blast will probably score you kills right then. At this point in the game it's now more important than ever to throw down your turrets where they can do damage, and keep your maximum range on grenade and rockets. You basically keep shooting in and out of range to fire things off as cooldowns get up.

Another important thing. Even with max cooldown reduction, your turrets will still be on the long side (particularly turrets and ult). You need to consider when a fight is likely to break out, and when is the most advantageous time for you to drop both turrets and turn on your ult. Generally I lay down 1 turret to do whatever, but always have 1 stored in case of emergency or team fight. DO NOT drop BOTH turrets while the teams are still posturing around for a fight. You will run the risk of the entire fight taking place without your turrets getting in on the action.... It's imperative that you drop your turrets at the right time, in the right place. In some fights that might mean dropping them right at the beginning to help immediately. In some fights that might mean holding onto them until the fight is initiated. If you drop both turrets too early, they will probably get instant primary and may still be on cooldown through most of the duration of the teamfight. Even more frustrating, you may lay them somewhere that the team fight moves away from.... Early Cooldown Heimer build is the only one I'd consider since it at least gives you SOME degree of flexibility in this regard. (YOU WILL NEED IT) 40% Max Cooldown at 18 is a must!!!!!!

It takes all three tools to get the job done. Turrets, grenades, rockets. You have to learn their nuances, and get used to their subtle quirks and annoyances.

You are a damage/support hybrid. It's a very odd role to have, but is similar to Lux in terms of team function. You exist to not die, pump out damage, and make chasing or standing and fighting an impossibility for the other team. You're also amazing at zoning and kiting in team fights.

Things start getting dangerous once you have your full items. Every minute the game goes on the enemy team gets stronger, and you get weaker. This build does afford you a LOT of damage, even in very late game... but the longer the game goes on, the more necessary it is for you to poke any enemies you can with rockets and grenades whenever the cool down is up as players are posturing. Every bit of health they don't have could mean kills and less deaths when the posturing ends and the fight breaks out. It's your job to poke, to frustrate and to never, ever die.

Your turrets are like skill shot shields... If you suspect a blitcrank grab, or an ahri taunt, or a morgana root, or a lee sin blue ball **** is coming your way. Be ready with that turret to block that **** as you're legging it!

Throw down a couple turrets in mid to deal with the upcoming wave, then gingerly walk off like you're doing blue. Instead make your way to either top or bottom for a gank. You'll have packed one turret into your storage at least by the time you get there, and the person in mid will likely be so absorbed in doing away with your precious that they won't think to follow... after all, they can't really leave them be and follow or else minions will be pushing their turret.

If you throw down both of your turrets at dragon, he will primary the first turret that attacks him. You can wait until the third breath hits the turret, then hit your ult to refill its health and continue tanking dragon. This allows for fairly safe, super early dragon steals.

Generally I will leave 2 turrets behind in mid and wait for at least 1 turret to pop back into storage, before I accompany the jungler on this task. This may or may not be possible depending on the level of play, but it will definitely be viable for some people in many levels of play.

You can actually place Heimer turrets over the wall into blue, red, wraiths, and baron if you're standing on the other side of the wall. This comes with its own dangers when you actually use it, but it can yield amazing results depending on the scenario.

I know I mentioned it before, but I'll mention it again since it's so important. USE GRENADE AS A SCOUTING TOOL. IT IS AMAZING FOR THIS! This is one of many reasons for building a cooldown / mana build. You can scout like crazy, then rocket when you see someone. (should you choose to do so)

When an enemy is trying to flee an engagement, you can flash behind them lay down a turret and ult VERY fast to slow them. A lot of times I hit fleeing targets with a grenade stun, then place turrets down and ult + rocket. By that time they are pretty much dead, and if they're not the turrets usually just finish them off while I kite around. Of course if it's a gank, they just get dps'd down and die.

In my experience, if your turrets are close enough to be shooting at the enemies near the onset of the fight, they're probably in a decent spot... especially if your team is in a position to fall back through them if things go south. Also you always hope that it's your tank initiating the fight, this makes it easy to sneak in real fast and plant at least 1 turret close enough to start blasting. This offensive turret plant can be super risky if you're not very aware of your surroundings!

If you're in a situation where you suspect you or your mate might be facing a dive, plant your turrets in line with or slightly behind your tower. This way they will serve as a second line of defense if they range the turret down with AD, or will totally frustrate anyone who's impatient enough to try and finish you with the turret still up. (ESPECIALLY if you have your ult up)

Sometimes I leave mid to hit a pushed top or bottom lane. I will roll in behind them and start setting up turrets so they have no escape, if they start trying to push through I just ult + stun. Remember, slowing them with turrets makes it way easier to stun them with grenade! Another thing I like to do (that's more dangerous) is to come around behind their front tower and double turret the lane so that minions can't get through to stop your friendly minion wave and friend who's pushing it. Then I just pop back and forth out of max rocket range firing until they die a lonely death on their tower. It's really evil, but hilarious so be on the lookout for when there is an opportunity to prevent their escape and lock them in. REMEMBER, SHUTTING DOWN AVENUES FOR ESCAPE AND ATTACK IS YOUR SPECIALTY!!

As an aside, I've found Heimer to be quite good in 3v3. His grenade, and turrets are extremely useful. He's good at clearing jungle by himself mid game, and very useful for stealing away to grab dragon early.