We normally say that we "keep a secret" or that we "keep (something) a secret." For example, "I will keep the secret" means "I won't tell anyone the secret." "I will keep your name a secret" means "I won't tell anyone your name."
"I'll keep you my dirty little secret" probably means that the speaker will not tell anyone about "you." It seems most likely that the speaker is referring to a relationship, and means "I won't tell anyone about the fact that I am in a relationship with you."
The speaker then says, "Don't tell anyone, or you'll be just another regret." "A regret" is something that you regret. The sentence implies that if "you" tell anyone about the relationship, then the relationship will end, and I (the speaker) will regret that we were ever in a relationship. You will be something I regret. (In other words, my relationship with you will be something I regret.)
(Another option is that the word "you" refers to the secret, as though it is a person. It could be that when the speaker says "I'll keep YOU, my dirty little secret," the speaker is talking to the secret, saying "I'll keep you." This was my first thought, but after reading the lines over, I don't think it's very likely.)
It's normal to say that you'll keep something a secret. It's a little more unusual to say that you'll keep /someone/ a secret, but in the right context it can make sense. Referring to a person as "a regret" is somewhat poetic, so people would be less likely to say this in conversation.