Was it the united states or the soviet union's fault for the cold war ?

There is no known reason as to why it started. I would say the soviet union is at fault because of Stalins desire to dominate the world under communism. 

Well that depends, do you stand by communism or capitalism. if you want to say it's USA's fault then I'd say it's because USA was trying to control earth without any intervention from other countries and that's what started this...
if you say USSR (soviets) that'd be because Stalin wanted to dominate with communism and his first attack is what started this...

or (if it's an option) you could say that it was an inevitable fight that was bound to happen either way, because both countries were power hungry...