It's a cold war, I'm a colder soldier
i.e. he is tough and strong enough to survive through all his problems.
A cold war is a state of political tension that does not involve the guns, armies, battlefields, etc, of a “typical” war. Common is saying he’s living in a state of tension in a rough life.
If you want to survive you have to be stronger than your enemy (being colder than cold).
To help improve the meaning of these lyrics, visit "A Dream" by Common (Ft. Will.I.Am) Lyrics and leave a comment on the lyrics box