"He doesn't seem to be rich" is the one that is common and would be heard in everyday conversation. "He seems not to be rich" is very poetic sounding and would really only be said in literature or poetry I think. It seems like it needs to be followed up with, "but rather only slightly wealthy" or something like that. If you just want to say that he doesn't look like he is rich, then say the first one.
The first is actually ambiguous, depending on whether it's the seeming or the richness that is denied:
(1) He doesn't seem to be rich, but he is really. [perhaps he looks quite ordinary]
(2) He doesn't seem to be rich, he seems quite hard up. [perhaps he's dressed shabbily]
Only (2) is equivalent to 'He seems not to be rich,' because in 'not to be rich' the negation is the clause denying richness. When it's in the upper clause, it can deny its verb 'seem', or it can reach down and deny the richness.