You can search for rental homes on realtracs.com - that probably has most of the rentals from many different property management companies. Also, MilitarybyOwner.com will have rentals for the area. We live off of exit 1 and really like the area. Plantation Estates, Fields of Northmeade, Franklin Meadows and Patrick Place are all nice subdivisions. You can search by subdivision on the realtracs site, by zipcode or just by Clarksville as a whole (but Clarksville is pretty large). There are nice homes in the Sango (exit 11) area of Clarksville but that means you would have about a 30min drive to post with traffic in the mornings. Exit 8 is the Rossview area and also has really nice homes but they tend to be a bit pricier. I would avoid Oak Grove, KY and the downtown Clarksville area (just personal opinions). Just in the last week, I've seen probably 10 for rent signs go up in our neighborhood. There are definitely plenty of houses available!
I would definitely stick to the Clarksville side of Ft Campbell. I was born and raised there, but PCSed back in March. We owned a home off of Exit 1. Like @JENandJEH said, that is one of the better areas to live. Those subdivisions she listed are great, as is Camelot Hills and Cornerstone (off Needmore Rd.). If you choose to live off of Exit 1, map the house and make sure it's at least a mile away from the dirt racetrack on Needmore. It gets pretty loud during the summer on Saturday nights because of it. Oak Grove just seems, more or less, existing only because of the military. A majority of the homes are older and seem to be run down. Also, utilities are MUCH cheaper in TN than they are in KY. We paid about $30 a month for water while my parents (in Oak Grove) paid $90 on average. You can PM me if you want more info.