I have rented a house and signed a 18 month lease with my landlord. My roommate moved in as well but did not sign the lease. My roommate and I had a verbal agreement that he would be responsible for his half of bills. In the past two months my roommate has been verbally and physically abusive. Disrespectful and I have 3 small children in the home as well. What process do I need to take to get him out of my house.

You could get a restraining order if you fear for your safety and that of your children.

To evict in Kentucky - "THIRTY DAY NOTICE: The landlord must give you notice 30 DAYS that they want you to vacate (move out) prior to the SUMMONS AND COMPLAINT BEING DELIVERED. (Most Cases this is the rule)
The landlord must inform you first of this intention. They may give notice by

Giving the notice to you
Giving it to any adult who lives in your home
Leaving it on your door. "

Kentucky Evictions