Disclaimer: This example is in Hawaii, check your states laws. Also this was a court auction; your situation sounds like an REO sale which is slightly different.
I recently closed on a foreclosure that the owner had rented out with no lease. I notified the occupants 5 months before closing (before I even won the auction - but this was an educated decision I made ONLY because of what I knew about the tenant) and they said they would love to stay, agreed to have the $3,200 on the closing date, and I updated them every couple weeks as we approached closing. On closing date they had an excuse of how they lost their job and needed 5 more days to have the money (how a family that has been living rent free for 2 years was unable to save the deposit and 1st month to secure their 3 children's shelter is BEYOND me, but they didn't). I gave them 5 days, at which time they said they needed 2 more, etc. I finally told them it was time to go, and they had to be out by the end of the week. That's when they stopped being nice, started getting loud and nasty and telling me that I need to give them 45 days (the length of time they would be entitled to if this were not a foreclosure).
Fortunately the lawyer that handled the sheriffs sale had filed a Writ of Ejectment, which gave me all rights to possession and gave me the ability to immediately have any occupants removed by the sheriffs department. I ended up paying a process server $125 to go over there and warn them that the sheriff would be physically removing them if they were still there when we came back. I was very lucky that it worked, because bringing the sheriff out would've cost me close to $2,000.
I would call some lawyers and process servers in your area to help determine if any type of writ has been filed on the property, and if not I would find out how much it would cost to get one filed (or go to the court house and see if the clerk will help you file it yourself).
My situation was best case scenario, I got them out in 2 weeks for $125... the next step would've been offering them $500 to be out by the end of the week (then $750, then $1,000 or have the sheriff do it). I was also lucky enough to (VERY CAREFULLY) convince them to pay their entire outstanding electric bill, which would've been passed on to me, before letting them pick up their last two large items that they left behind. But that can be risky depending on your states laws and the occupants knowledge and awareness of those laws. I probably wouldn't risk that in most situations, but in this case I knew it would work.
Regarding a tenant that already has a lease in place: yes they have the right to stay and you have to abide by the lease (check your states laws to confirm, but this is probably true in all states). BUT, the tenant may not know that, SO if you can convince them to agree to move out WITHOUT forcing them to... that could be a possibility...some people won't want to stay once they realize they are not wanted.. just be careful and make sure you have a way to prove for yourself that you did not force them out... get something in writing saying that the tenant is choosing to terminate the lease maybe?
At the end of the day, I bought this top floor renovated ocean view corner unit for 40% below market value in the hottest market since '05, on the highly desirable island of Maui, and had the occupants out in 2 weeks and had my new tenant moved in and cash flowing by the 3rd week after closing. I'd say I got pretty lucky, combined with my extensive due diligence of course. I hope your deal goes just as smoothly as this one went! Good luck!