Okay, so before everything, I'm just a noob (lvl 11 at the moment), I'm just trying to make some good build for champions I'm probably gonna be playing and trying to make a decent team out of it as well.

As you can see I haven't selected any runes, cause I simply never used them and don't really know about them.

also please read my request

Items

So this is something I think will work.
First buying Doran's Ring because it's cheap and very iseful to anivia in the beginning, then at the end you sell it for Deathfire Grasp if the game lasts so long.
After Doran's ring you really need the boots because of the little helath you have and one of the slowest base movement speeds.

What I know is that Anivia almost entirely relies on her abilities, so you will need a lot of mana (and AP ofcourse). That's why we get Tear of goddes after the boots, especially because it will keep your mana pool growing with time until a maxium of 1000 mana.
When we've got these items other champions will surely also have boots, so we need the second enhancement and the magic penetration is also a good thing cause we rely on our abilities.
As the stacks of tears of goddes stay when you upgrade it to Archangel's Staff, we'll first get Rod of Ages, because it needs time to get to its maxium potential.
Ending with an Amount of 3330 mana
After this it's mostly AP staking ending with 410*1.30=533 AP
This also makes Deathfire Grasp even deadlier with it's active instead of 30% it does 47.5% magic damage of somebody's current health

Skill Sequence

Not you're most common sequence but I think it's a good one, When becoming lvl 5/6 you will start getting strong enough to solo lane and really push, maybe even being able to 1vs2, You can start harassing people :). So when you become lvl when can use Crystallize to scout bushes, give you just those 1 or 2 seconds extra to either make a kill or getting to your tower before getting killed. Also at lvl 5 you should have enough mana to use it once in a while.

We put a point in flash frost cause it multiplies the damage output by 1.5 and it reduces the cooldown, the extra mana at lvl 7 shouldn't be a real problem.

Frostbite is of course your main damage output, because a 5 sec cooldown, low mana cost and great damage, just AWESOME skill.

Items

Normally you would start with Doran's Shield to make you more durable and better able to harass, but if you see that the enemy champions you're are playing against aren't annoying champions, buy a Saphire Crystal and 2 or 3 Health pots.
Sheen is one of the most important items for Blitzcrank. Because it gives you insane burst damage in combination with you Power Fist. After that the item that you should be getting depends on wether you're winning or losing/"being harassed", if you're on the winning hand get A phage and upgrade it to Triforce otherwise get a Banshee's Veil. Then get that item of the 2 you didn't get yet. After that you normally go for Randuim's Omen, but if you really need the extra damage get the Infinity Edge first.

The mana items we got are certainly useful, it gives you a bigger shield in combination with you're passive and you'll need it for the Rocket Grab and other abilities

Skill Sequence

As well not you're most common sequence but I think it's a good one. When you get sheen you should be around lvl 5, at that point the cooldown of Power Fist starts to matter. Before we put 1 extra point in Rocket Grab because, that will double the damage and decrease the cooldown. From that point on our AD and Power fist start to become really powerfull so after Power Fist we upgrade our attack speed and our escape/chase ability. Always putting a point in our ultimate :).

Another possibility is to start with putting a point in Power Fist first and then one in Rocket Grab, this is good when as summoner spells you've got Flash and another. And you don't wanna drain your mana like hell in the beginning. This totally depends on the situation.

Summoner Spells

Team build

Okay so basically you have Anivia as AP Mage burst, Karthus as AP Mage burst/R-Dps (lay waste 1 sec cooldown), Nasus as AD M-DPS, Blitzcrank as AD M-DPS/Beefy (a little bit tank) and Shen as Tank/little bit jungler.
To me this seems pretty well balanced, especially if you consider that you don't need a support, because all of them have CC/support skills, Anivia: slow, stun and crytallize, Karthus: Wall of Pain, Blitzcrank: Rocket grab and Power fist, Nasus: Wither and last but not least Shen: Shadow dash, Stand United and you could consider Vorpal Blade as a support (healing to all champs who attack certain enemy).

Positioning

Possibility 1:

Mid-solo: Karthus

One of the best for mid-solo can out-zone almost anybody

Top-lane: Anivia with Blitzcrank

great combo

Bottom-lane: Nasus with Shen

Make sure Nasus gets last hits with his Siphoning Strike this is very important, however it has a 4 sec CD, meaning he doesn't have to get all the last hits.

Possibility 2:

Mid-solo: Karthus

One of the best for mid-solo can out-zone almost anybody

Top-lane: Anivia with Shen

If Anivia needs better protection, its sometimes a better choice to set her up with shen.

Bottom-lane: Nasus with Blitzcrank

Make sure Nasus gets last hits with his Siphoning Strike this is very important, however it has a 4 sec CD, meaning he doesn't have to get all the last hits.

Possibility 3:

Mid-solo: Anivia

better against some champions then Karthus, I would prefer to put Anivia against Vlad instead of karthus

Top-lane: Karths with Blitzcrank

Very deadly indeed

Bottom-lane: Nasus with Shen

same as before

Possibility 4:

Mid-solo: Anivia

better against some champions then Karthus, I would prefer to put Anivia against Vlad instead of karthus

Top-lane: Karthus with Shen

same as before

Bottom-lane: Nasus with Blitzcrank

same as before

Other possibilities:

I don't really recommend this but sometimes, it could work out better to set Nasus with Anivia, especially if the other team keeps harassing Nasus with ranged Champs.
It also is a choice to protect Anivia the first 2 to 6 levels or so with Blitzcrank or shen. After that blitzcrank would go back to shop to buy items and then goes bottomlane helping Nasus while Shen is jungling. It could be necessary to set blitzcrank in the bottom-lane from the start if it would be to hard for Nasus and have Shen help Anivia/Jungle.

What if you've to play against Vlad, Then it might be a good idea to let Anivia go mid and when she and Vlad are lvl 5, either shen or blitz should help her take out vlad. At lvl 5 she can use crystalize to counter his pool and Blitzcranks rocket grab goes through terrain if I'm correct so it can pull vlad back behind her wall. Shen's taunt also goes through the wall.

I really have one favor to ask, comment, criticize, but pls explain/base your opinion.

I wanna get better and I want to learn, but I will need your help.

I don't mind if you say it suck as long as you can give me a good explanation why.

Also don't go saying anything about it not being complete/finished yet, cause it just isn't and probalbly will never really be finished.

I would really appreciate any tips, recommendations, help and critticism.

Thanks for the reading my Build and thanks in andvance for any help.