Is there a way to watch a movie purchased through iTunes from my iPad on my AppleTV?

Yes, indeed there is.

Ensure that airplay is not disabled on your Apple TV and that both devices are on the same network. Then when playing the movie on your iPad, select the airplay icon in the control bar, tap it, select the Apple TV from the contextual menu that appears. The movie should now stream to your Apple TV.

If your Apple TV is logged into iTunes using the same Apple ID that was used to purchase the movie on your iPad, I believe you should be able to see the movie for streaming over the Internet in the Movies/Purchased section of your Apple TV menu. Assuming you have Internet connectivity of course. If not, then using Airplay to stream the movie from your iPad makes sense.

This is no way to tell for sure, because this is a hit & miss. Some movies you can do this, other times you won't get away with it. Generally, consider yourself lucky if you get a RENTAL to work this way. Purchases will probably work though.

To be honest, it's always best to just purchase/rent a movie through iTunes on your computer first, this avoids many problems.