Harry accompanies Dudley, Piers Polkiss, Aunt Petunia, and Uncle Vernon to the zoo, where he unknowingly uses Parseltongue to speak with a boa constrictor. When Dudley notices and comes running, the glass of the snake’s enclosure mysteriously disappears and the snake escapes.
This is a Saturday according to the book. It comes approximately one month before Harry's birthday. That morning, Dudley receives a cine-camera for his birthday:
"At that moment the telephone rang and Aunt Petunia went to answer it while Harry and Uncle Vernon watched Dudley unwrap the racing bike, a cine-camera..." (PS2)
After the zoo incident on Dudley's birthday, Harry is confined to his cupboard for a few weeks. By the end of that period, the cine-camera is broken:
"By the time he was allowed out of his cupboard again, the summer holidays had started and Dudley had already broken his new cine-camera..." (PS3)
Then, a week before Harry's birthday, on the 24th of July, Harry receives a letter addressed to him in his cupboard under the stairs. Uncle Vernon moves Harry to Dudley's second bedroom upstairs, where Harry sees the cine-camera and we learn that Dudley's birthday was a month ago:
"Nearly everything in here was broken. The month-old video camera was lying on top of a small, working tank..." (PS3)
We know from the book that July the 24th was a Tuesday. The Saturday approximately one month before Tuesday, July 24, would be Saturday, June 23. So we know that Dudley's birthday is June 23. Like most dates in the books, this doesn't correspond to the dates in any particular "real life" calendar. (PS2)