I have dual citizenship, USA & a Caribbean island. If I enter the USA with only my Caribbean island passport, will they know I am a USA citizen?
As a US citizen, you must enter the USA under your American passport, not using any other, it's illegal to do otherwise.
I am an American citizen, but also have a foreign passport. Can I use my foreign passport to travel to the United States?
No. Under U.S. law, U.S. citizens must be in possession of a valid U.S. passport to enter or leave the United States. This is true even if you hold a passport from another country. If your U.S. passport has been lost or stolen, or if it has expired, you must apply to replace it before traveling to the United States.
Source (US embassy in the UK)
U.S. nationals, including dual nationals, must use a U.S. passport to enter and leave the United States.
Source (State Department)
As for if they would know, that depends (there are ways that suggest it, including but not only place of birth, along with some databases), but Travel.SE policy is not to aid people in breaking the law so alas all we can say is you need to enter on your US passport...