This means gambling at casino table games, such as roulette, poker, blackjack, etc.

The answers you've received are correct about the meaning of "play the tables", however your sentences(although perfectly grammatical) aren't accurate, unless they're not meant to be about the film. It's not illegal to play the tables in Monte (Carlo); Monte Carlo is a famous casino and tourist attraction in Monaco. To be caught doing something suggests that it's something you shouldn't do, so unless the character wasn't supposed to be gambling, it's a strange thing to say in your composition. If you're just trying to use the phrase "play the tables" and it has nothing to do with the film, then you're fine.

'Play the tables' means gambling in a casino. Most casinos have different tables around which people play games like blackjack, roulette, craps and poker. You could say "In some states, playing the tables is illegal", but I think it sounds more natural to say "...gambling is illegal". You could say "He was caught playing the tables in a casino in Las Vegas" I hope this helps. Emma - Talk2Me English