The place looked great and I listed it for lease for $1,375.

Get a copy of the rent rolls and leases.

When I toured the unit before purchase, the tenant keep the place very clean and no issue beyond nonpayment appears to be going on as far a lease violations.    

The current tenant started lease in March of this year.I had installed new carpet: $1200, new furnace, $2,221, and new water heater: $950 all this year.I plan to have the property for 3 to 5 more years.Thank you in advance for your help, I really appreciate it.

How do I handle the rehab with renters in the house or is it advisable to let their lease end and have them vacate before I take on the rehab (since the additional bedroom and bath would be for a new tenant anyway? 

Tell them to open the door.Wow... your lease says you are there for them to abuse?  

This lease agreement holds the full weight of a written month to month agreement, without providing you any of the protections given to you under a written lease.

You can charge them a rent and sign a lease for 30 days max taking a deposit.

You'll have to honor the existing lease agreement, unless the tenants want to voluntarily leave or you opt to offer them cash for keys to get them to move out once you take over ownership. 

I'm guessing there is no lease so this situation is a little more complicated than a typical eviction which is generally because a tenant isn't paying rent or abiding by terms of a lease agreement.