This is one of the hardest minigame achievements. 25 points really is a very, very high score in this game and it took me a while to get the technique right to achieve it.
The problem is the time limit is extremely mean so you feel that you have to throw fast. Throwing fast means you don't have long to aim, and a lot of the time you'll find that your throw goes wildly wayward. You can't really afford to miss more than once, twice at a push.
So, here are my tips:
Refine your throwing technique. If only for this minigame, you must avoid any backswing in your throw. It is the backswing that seems to either radically alter your aim or even "cancel" it altogether. So when you're aiming, your arm should be bent at less than 90°, your hand no more than 5-6 inches in front of your face. Get your aim quickly, pause momentarily to set it (less than half a second or so), and then throw by moving your hand in a direct line forward.
You might also find it helps to use a tight fist rather than miming the throw of a dart with an open hand. Perhaps the sensor can track your hand with more accuracy like this.
If you feel you need to tweak your aim as you throw (the balloon is moving or your aim was a little off) you can compensate as you throw by moving your hand in the direction you would like to adjust in. Throw at an upward angle for example to make the dart go slightly above where you aimed.
When you have moving balloons, take extra time aiming. Aim just ahead of the balloon's position rather than exactly where it is, because they move quite fast. Compensating for it's movement in your throw is also a good tip for these. Take extra care though, because some of these are very small, and very close to the mascot. Hitting the mascot results in a 3 second penalty and that's game over as far as this achievement is concerned.
When you have static (non-moving) balloons, you can afford to speed up your throw. Aim quickly, throw quickly. You will get used to the rhythm that the game can cope with after a few tries. Despatching 3 static balloons on one board should take less than 2 seconds. This is where you need to make up time.
Using those tips, this is not as crazy a target as it may have seemed. I was stuck on 13 points for ages. I got to the dizzying heights of 20 once, and it had my heart pounding. 25 seemed impossible. But by altering my throwing style, controlling my breathing, concentrating, slowing down for moving targets, and speeding up for static ones, I have managed 26. And it would've been 27 or more if the achievement popping had not put me off my throw.
So keep trying and good luck!