If you're like me, you may have had difficulty finding where to make purchases. It's under the Customisation option in the main menu.
Pick any character and then select one of the following categories: Shirts, Pants, Shoes, or Accessories. Scroll down with the D-Pad and you will find items for purchase. Select item, hit confirm, and you've made a purchase. Continue to do so until you've spent $250,000 and the achievement will pop. You can purchase items on any character as the items are all shared.
Note: I suggest starting with accessories because after you make a purchase the game keeps you in that category whereas others boot you back to the category selection menu which is a minor annoyance.
How to Make Money
The only way to really make money in this game is to win Fast Money on Easy difficulty against the Lee family. Each win will earn you $20,000 so you are looking at 13 wins to have enough to get the achievement. If you lose Fast Money then the only cash winnings you receive will be your total point equivalent in dollars.
Also note that Battle Mode isn't really a good option for this because if you win Fast Money you only get 10x your points in cash.
There is no bonus cash for winning multiple games in a row. You will get the car if you win five games in a row but there is no cash bonus associated with doing so, so just keep grinding out those wins.