I like him working out.
In terms of grammatical case, him is marked accusative. If you want to get into syntactic theory, then it is involved in something called Exceptional Case Marking.
This Wikipedia article is a bit confusing, even if you know what ECM is. Essentially, [him working out] is analysed as a clause, so the sentence is structured [I like [him working out]]. So, [him working out] forms a grammatical unit. The Exceptional Case Marking is the fact that him is getting accusative case as a result of the verb in the main clause, even though him is in the subordinate clause (this normally doesn't happen). So him is serving a dual role in this sentence, according to theory.