Doom would actually be fully capable of becoming an arch nemesis to Superman, sort of like the way Lex Luthor is. Doom has money, intellect, all the resources he'd need. He could easily get so obsessed with the Man of Steel that he'd want to find a way to steal his powers or destroy him. Remember, Superman is a boy scout. He wouldn't just walk up to him and punch a hole through his face or something, he'd have to find a way to beat him within his own moral framework, just as he does with Lex.
If you mean a straight up battle royale, then Superman takes him in .5 seconds.
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