This can be done on the first trip to Gold Saucer. Go to Wonder Square, and if you haven't recruited him already, Cait Sith will join the party. Continue up the stairs and across the walkway into the games area, then cross the room into the second area. There's a guy on a motorcycle game; go counter-clockwise around the room to get on the game next to him. Each play costs 200 gil.
This is a repeat of the Escape from Midgar scene, only using points instead of preserving HP for a boss fight. The controls are the same,
to move, and
to swing at the respective side. You get 500 points per bike destroyed, and 50 points lost every time the van is hit.
The bikes are color-coded to difficulty, with blue being the easiest, orange intermediate, and red as the hardest. Red bikes will actively dodge your sword swings, so your best bet is to ram them. Ideally, push from the side into a wall or get ahead of them and brake. Otherwise, they will be frustratingly hard to kill and the next wave won't spawn until they're taken care of. Blues and oranges can be rammed or slashed, with ramming typically prioritized if they're threatening the van.
Destroying a bike is Once you can fairly reliably hit the bikes and ram the reds, the achievement shouldn't be too bad but may require extra attempts depending on skill. The game is two minutes long and on a decent run, you should expect to pass 10,000 points around a minute and a half.