I've been Dating this guy for a few months now. He claims he's divorced but all signs are pointing otherwise: he turns his phone off at night, never invites me over his home and always wants to take me out of town on business trips. How should I break up with him?

Tell him that you think he is married, and tell him why. Tell him that it is over unless you go to his house NOW ( wait till he is with you to tell him that) Do you know his address ? I would perhaps just show up. Say surprise and see how big of surprise it is.

You tell him it is not working. You tell him you are not comfortable with the fact that he turns is phone off at night, he does not invite you to his home, that you feel like a mistress.
Unless he can prove you wrong, you're done.