You have the right idea; an isomorphism classes partition the set of all relations on X, where two things are in an isomorphism class iff they're isomorphic.
What does it mean for two relations to be isomorphic? There's a natural model-theoretic definition of an isomorphism. If you have a language where is a binary relation (the upper bound you gave gives me the impression we're only interested in binary relations. Doesn't really matter in ZFC though, as there are just as many binary relations as n-ary relations).
Two L-structures M,N are isomorphic iff there is a function bijective such that , where is the interpretation of the relation R in the model M.
Sorry I was long winded. I'm not sure if you were asking for the formal definition of isomorphism, so I thought I'd be safe and state it.