You just need to get your available total to 25,000 gold at any point.
The achievement wording might be interpreted as saying you need to spend zero until you reach 25k, but that's not the case. It just means it's your current balance that matters, not your all time total earnings.
So buy the early upgrades to make things easier, and wait till later in the game to save up. You can replay levels too to give your total a further boost, if you're impatient like me.