I don't think there's a tri-club there, but Nashville has a large and active club. It's only about an hour from Clarksville--
www.gnac.org .
There are plenty of races within a reasonable drive of Clarksville. The first one of the year is Memphis in May oly. There's a few sprints in northern Alabama/southern middle Tennessee, a Tri-America oly in Louisville, oly's in Nashville, Knoxville and Chattanooga and a 1/2 in Knoxville.
I've never ridden around there, but I would suspect the riding would be fantastic 15 minutes out of town. Nice scenic, rolling terrain and quite rural in the countryside.