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US citizens (dual or not) are required to enter and leave the US using their US documents, in general a US passport. So technically it's illegal, although it's not clear what the likelihood of getting caught and the penalty would be.
In general a US consulate or embassy will NOT issue a US Visa to a US citizen, so you'd have to lie on the application to get one. I wouldn't recommend that.
Your best course of action would be to apply for a US passport.