Sounds like an adjustment problem. There are excellent instructions on how to set up and align the DR on the Park Tool site among others.
First and easiest way to check without taking anything apart is to shift to the highest (smallest) cog and check the cable tension. There should be almost no slack in the cable, but it shouldn't be overly tight either.
If there is any slack, use the adjuster to snug it up.
While you're back there, eyeball the DR' arm, the hanger, etc and make sure it's dead straight. It's easy for a slightly bent hanger to cause similar problems.