I made this build after I played a few matches with Nunu and stumbled across this. It is a melee attack damage/speed build for Nunu that worked extremely well for me. In my first pvp game with him I went 18/2/9 and slayed the most minions and turrets of both teams. Enough of how I made it and on to how I would suggest playing him.

1.1
Moved points from Strength of Spirit to Perseverence
Fixed ability tree

In the beginning of the match I take Ice Strike first because of the decent damage and the very high slow effect that will allow you to destroy casters and people fleeing. I play the first few levels very aggressive by harassing the enemy with Ice Blast every time its off cd. At level 2 I take Consume because of its healing effect which you will probably need by then if you are aggressive. I use it every time its off cd and my health is at/below 3/4. At level 3 when you get Blood Boil I only use it when I attack somebody but this will change once you get the Chalice of Harmony. The reason I got this is because it fixes Nunu's mana issues. After I get the Chalice I then keep Blood Boil on at all times. Keep doing this for the first 10 levels and you should be good.

Consume: I use consume to either heal or help kill monsters. I use it every time it is off cd and my health is below 3/4
Blood Boil: I use this ability to chase enemies and inflict more damage. After you get the Chalice of Harmony I keep it up at all times because of the move speed. It is great for destroying towers fast.
Ice Blast: I use this to slow enemies to chase and kill them. I usually use it as an opener to make sure they dont try to flee etc.
Absolute Zero: I don't use this ability alot because it usually gets you killed or does too little damage to be effective. I would use this to kill minions.

I buy long sword first with a health pot because you will be at low health by the time you hit level 2 if you play aggressively by harassing the enemies with Ice Blast. The first item I build is the Chalice of Harmony because it fixes all the mana issues I have with Nunu. Next is Berserker Greaves for the move speed and attack speed. Next I get the Sword of Occult because it will give you 100 damage and even more move speed at 20 stacks. Next is Sword of the Divine it gives you a grand 60% attack speed and complements that by making you deal 100 damage every third attack. Next is Bloodthirster which will give you some attack damage and turn all that damage into health. The final item is Madred's Bloodrazor which gives you damage attack speed and armor and will complement attack speed by dealing 4% of the targets maximum health in magic damage.

Pros:
A lot of damage
A fast attack speed
A very good slow
Can destroy monsters including dragon and baron
Can take a lot of damage
Fast move speed
No mana issues
Heals fast
Cons:
Vulnerable to CC's (countered with cleanse)
Not much ranged power