Ten years ago, there had been lots of pictures of what looked like a large pink beach ball wearing different-colored bonnets... (PS2)
Dudley is the only child of Vernon and Petunia Dursley of Privet Drive, Little Whinging, Surrey. He is about about a month older than his cousin Harry who comes to live with the Dursleys in November of the following year. Dudley grows up spoiled while Harry grows up neglected and downtrodden.
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.