How do we know when Dudley's birthday is? Well, we know that Harry's is July 31. A week before his birthday in 1991, on July 24, Harry begins getting letters from Hogwarts. At this point, Harry is given Dudley's second bedroom, where he discovers Dudley's "month-old" video camera, broken (PS3). That camera was given to Dudley for a birthday present (PS2) on Dudley's birthday, which must therefore be about a month before July 24.
No, wait, there's more. We know that in the year Harry turned 11, Harry's birthday was on a Tuesday (PS3) and Dudley's was on a Saturday (PS2). Counting back a month from Tuesday, July 24, we find that the nearest Saturday is the 23rd. So regardless of the year in which this is taking place, Dudley's birthday is almost certainly on June 23.