どうにか

どうにか means "somehow": something got done, or has to get done, though it was hard and there was no clear way. It is an adverb and goes before the verb. Said of something past, it means "just about managed".

どうにかなる and どうにかする

With なる, the situation sorts itself out: "it'll work out somehow". With する, someone does something about it, and in a request it asks for help. どうにかして before another verb means "by some means, one way or another".

どうにか or なんとか

なんとか means the same and can replace どうにか in every use above, and in conversation it is the more frequent of the two. The pairs match: なんとかなる, なんとかする, なんとかして.