*As I do not play Ryze, Veigar, or Vladimir that often their sections of this guide are lacking to say the least. I have attempted to fill them out based on my prior game knowledge and a few games I have played to test out some theory for this guide. Their mastery/rune/skill order/item builds are subject to change and I will be reading the comments for advice. I know the roll that they should fill on this team and how they play, just not the best way to build them. A downvote given because of any faults in these characters' builds would be a little harsh seeing as this is a guide for how to play using this team composition, not a guide for the specific character.

*I have attempted to avoid using snowballing items in this guide due to their inherent risk that they provide. This team composition is designed to provide the most stability possible regardless of luck and misfortune. Mind you that if your team pulls an early lead than grabbing a Mejai's Soulstealer on Katarina is highly advised.

Welcome to the ideal team that I have brewing in my mind. This has mainly been built off of theory though I have played with 3-4 of these champions on a team here and there so I'm quite confident in the strength of this team.
Speaking of the strengths here's a brief look at what this guide and team composition will provide:

Pros:
-Key roles covered by, what I feel to be, the strongest champion to cover them
-A detailed guide to lane phase
-Minimal risk
-Covering of individual character weaknesses through the use of well paired champions.
-Versatility, don't fear your opponents' team composition countering yours. You're covered!
-Staying power: Don't get caught having to return to base frequently!
-Strong pushing power: Maximize your turret damage!
Cons:
-Somewhat magic damage heavy. I have attempted to cover this as strong as I can but magic resist is this team's primary hinder.

*The pros and cons lists will be updated when I see faults during play. Feedback will also be considered in this list.*

My ideal team composition is team one, but I've provided 2 alternative champions to take (if you take one, you should take both to cover the weaknesses exposed) in the case that your players are more comfortable with these champions. The alternative champions are also more commonly owned and cheaper to buy with IP if unowned.

I find that team 2 lacks a little on stuns, which I tried to make up for by providing Ahse's group slows and her single target stun ult. Aside from this minor change you should be just as versatile using this team composition as team one's composition.

*All lanes refer to the buffs on your team's half of the map*

Blue buff lane:
Rammus
and
Caitlyn or Ashe
Reasoning:
Rammus is probably your least gold hungry champion on this team while Caitlyn/Ashe is going to rather demanding for gold so they can rush their main damage item(s). Thankfully Caitlyn and Ashe both have a reasonable pushing power and staying power so it's okay for Rammus to just idle near the fight to get experience and carefully get some last hits. I find that Rammus is a quite mana hungry champion early game so keeping him near blue buff gives him the option to grab it and restore his mana rapidly, and having a reduced cooldown on his taunt doesn't hurt.

Mid Lane:
Vladimir
Reasoning:
See the chapter on Vladimir.

Red buff lane:
Katarina
and
Veigar or Ryze
Reasoning:
This is your "kill lane". This lane will focus mainly on denying the opposing team experience. Katarina should be using her daggers and Shunpo to punish champions that get too close to the minion line while Veigar or Ryze should attempt to keep control of the bushes to scare the enemy champions even further back, and make proper use of their stuns + nukes on any champion that dares to venture too close. With proper positioning Katarina should be able to Shunpo to the opposing champion at this point, or at the very least to Veigar/Ryze to cover some quick ground and join in the fight faster. This lane should play rather aggressively and start going for kills as soon as they hit level 4.
I would also like to note the importance of last hitting minions in this lane. You want to keep the minion waves as close the break between the bushes as possible so that you have the whole bush to venture into and push the enemies back (denying experience and last hits) and also providing you with some ground that they will have to cover if they need to flee to the safety of their turret (providing you with more space to safely damage them).
*Side note* Remember that Kat can go grab Lizard buff any time to provide her with a nice slow until she gets her Rylai's Crystal Scepter.

Welcome to the role of playing the main tank for your team. As with all tanks, your role is to keep the damage off of your squishy team mates, in the case of this team your main concern is your ranged DPS and your nuke/rooter.

Why did I pick Rammus for this role? Simple, his threat. To put it simply, there is a large reason to be worried about an opposing Rammus. Rammus can cover ground fast, reflect your damage back to you, lock one of your team members down with his taunt and cause some nice AoE damage (which can also be used to damage turrets). If Rammus goes ignored in a team fight he will cut a team down one player at a time.

Since you will be focused I have determined that Cleanse is an invaluable asset. This one skill will remove most of the CC that will be thrown on you at the start of a team fight, allowing you to proceed with your rampage through the opposing team. This will also mean that if your opponents want to shut you down they will have to expend more skills on you and have less to deal with your team, which should be supporting you heavily at this point.

Fortify is mainly a toss away skill but with the provided mastery set-up early game it should help you get last hits and cover your lane if things aren't going your way. Later in the game, if you can catch someone attempting to back door your turret, you can charge in with your power ball, get that tiny stun then proceed to activate fortify to double your turret's damage output and taunt your opponent to keep them in tower range. This should amount to a bonus 600 or so damage and net you some easy kills.

You may have noticed that I have Ashe and Caitlyn set up the exact same way, even down to only the first 5 purchases for their items. That's because their items are what make or break your team composition, these 2 characters are the most versatile damage dealers that these 2 team sets contain. We will be using this versatility to specifically counter the enemy team's entire composition.

You will be building with damage as your number one priority, no questions asked. So these are the items that you should be looking at the most:


Your first thing to consider is who the highest threat target is on the opposing team. This will usually break down to their carry or tank. This will be your main target to build against. Here are a few examples of high threat targets and how to build against them.

Ranged DPS or Mage
ex.
Kog'Maw
Karthus
Build a Bloodthirster then an Infinity Edge

Hp tank
ex.
Cho'Gath
Build a Madred's Bloodrazor then a Bloodthirster

Armor Tanks
ex.
Rammus
Build a Black Cleaver then a Bloodthirster and lastly an Infinity Edge

Idealy you will be building against the opposing tank primarily, allowing Katarina to focus on the opposing carries but in some cases the opposing tank doesn't provide enough of a threat to your team to really go noticed. You have to be the judge of when this is the case, but I've found that there are a few common tanks that don't worry my team that much.

There are:
Singed
Shen
Sion
Taric
Alistar

The reason that I find these tanks to be non-priority targets is because they have most of their effectiveness come from being focused, or they just don't provide enough offence to be a threat. For example, in the case of Sion he provides his team with a single stun and some health regeneration if they stand near him during his ultimate. Okay so one person will be stunned, probably Rammus who will just use cleanse and keep going, and some melee DPS could heal. The priority here is the melee DPS, Veigar should use his stun then nuke the melee DPS hard enough to kill, or almost kill it, during which point Sion should also be caught in the stun so he won't be able to get any hits, which will lead to no AoE heal at the time.

Take your lane! Vladimir is probably one of the strongest champions to take the solo lane. His built in life steal provided by Transfusion will allow you to stay in the lane for as long as you want, and provide a great harass against the opposing champion. You should be able to tower dive at will by pushing the minion wave up and using your Sanguine Pool to lose turret focus and allow you to scare off or, better yet, kill the opposing champion and land free hits on the turret. You want to break this turret as fast as possible so you can move into the ganking phase. If you think you will need help getting the kill call on Rammus to Power Ball from the bush and use his stun, they won't know what hit them.
*Note* If your blue lane is holding fine with Rammus in middle take the oppertunity and have Rammus hit the turret with you, likely destroying it if the two of you average at level 6.

*Note* My inexperience with Vladimir greatly hinders my ability to go further into detail in this section of the guide. I will be watching how my regular team's Vladimir plays so I can write some more relevant information here. For some general playing tips there are other guides on Mobafire which greatly cover playing Vladimir.

This team will be using Katarina as high survivability and high damage character, who's goal will be to focus the opposing team's main damage and support champs. We already have Caitlyn/Ashe focused on countering the opposing tank so you should be able to disregard the tank as a whole. Keeping this in mind we need to look at building a strong Katarina. Our masteries are designed to put emphasis on being able to survive if any opposing champ focuses you, but you will still be able to dish out a ton of damage.

If your team is faced with a team stacking large amounts of magic resist, or if you just find that your team is in dire need of more pushing power it is up to you to switch off of the comfortably of AP and switch your build over to physical DPS. In the item list for Katarina I provided only your starting set and your ideal boots, below is a list of useful items for AP and AD builds. Which path you choose is entirely subject to how your opponents build and how your team is performing. These items are listed in descending order for your standard purchase order.

AP
Rylai's Crystal Scepter
Hextech Revolver
Rabadon's Deathcap
Hextech Gunblade
Void Staff
*Note* Mejai's Soulstealer can be purchased at any point if you are feeling confident. I advise to get it at the latest just after your Rabadon's Deathcap or it may be to late to generate enough stacks to use it to it's full potential (aim to maintain at least 14 stacks). If you don't take a Soulstealer I'd advise taking a Banshee's Veil to give you some more survivability because you'll need to stay in fights longer.

AD
B. F. Sword
Phage
Bloodthirster
Frozen Mallet
Black Cleaver
B. F. Sword
Wriggle's Lantern or Zeke's Harbinger
Bloodthirster
*Note* If you feel confident you can buy a Sword of the Occult after your phage and not buy a Lantern or Stark's later.

*Note* I have specified the use of Clairvoyance in your build so that you may act as a scout for your team. Use this frequently for map awareness and to check for ambushes. Keep in mind that the enemy team can hear and see that you have used this, so you must be ready to act fast and not give them time to prepare for your team's attack.

The champion filling this slot will provide your team with a strong magical nuke. What I like with these champions is that they provide a nice root/stun with their spell chain. I would like to have Ryze in this slot in my Team 1 composition because of his more continuous DPS but he lacks in CC compared to a well played Veigar.

Veigar
This champion is amazing for his nuke, I can easily get around 800 AP by mid game and over 1000 AP by late game, and with his AP scaling this really hits hard. Mana management problems are a thing of the past thanks to Veigar's passive, essentially acting as built in Chalice of Harmony. We focus on getting a Rod of Ages and archangel staff to add 3 things to our Veigar
1. Survivability
2. Ability Power
3. Large mana pool - The more mana you have the faster you can cast spells, even when running dry because you should be gaining 200 mana in about 10-12 seconds. This rapid regen also attributes to you being able to stay in the lanes longer and not have to warp back to base just to refresh your mana.

Ryze
There's nothing fancy about Ryze, which is something I like about him. He has three moves that each can deal considerable damage and can scale fairly well with AP, but that can apply to a lot of champions right? Right. Here's what sets Ryze apart from the crowed: his passive. When a spell of his is used it reduces the cool downs of all his other spells, this can lead to just being able to toss spell after spell at the enemy.
Let's take a quick look at his spells:
Overload: Look at this, damage based on your mana pool, scales well with your AP AND has a relatively low cool down? No downsides here, just pure damage.
Rune Prison: We only put one point into this for it's root. It deals a fair amount of damage but is a lower priority than his other spells for the most part.
Spell Flux: Our main priority spell, this will keep minions pushed, enemies harassed, get you easy last hits, and when used right can put out heavy damage on a single target (just get them alone, pop a Run Prison on them and let your spell flux bounce back and fourth between you two).
Desperate Power: Nothing too special here. Use when the enemy team is clustered to get some nice AoE damage going on and also to quickly clear minion waves when pushing.

I don't want to go into too much detail about your casting sequence but here's what you need to consider:
1. Do I need AoE or a bit of a buff? If yes: Use Desperate Power
2. Root 'em in place with Rune Prison and get a good Spell Flux out on them.
3. Overload for some major damage and to lower your cooldowns.
4. Repeat steps 2&3 when available in any order.

Remember, you are easy to kill as either Veigar or Ryze. Stay in the back of team fights and watch for ambushes!