The quickest means of obtaining this achievement comes in local versus using two controllers. Set it up so that you are using any character you feel like with a 25% health handicap, then make sure you set the AI to a handicap of 0%. It's also suggested that you increase the rounds to best of 7 in order to reduce load times. Once the match starts up, double tap down and press your medium kick and punch buttons. Standard controls this would be triangle and circle, this activates Pandora. Attack the enemy an instantly win, repeat the process 300 times and you'll have your trophy!
This process can also be done while playing against an AI, that's why you'd want to keep your health at 25% just in case they retaliated. When using Pandora your gauge also immediately refills, so if you would like I'd suggest getting your 500 ex special moves and 100 quick combos while going for Pandoras. Naturally, if you're going to do this it's recommended you'd up the health of the second controller/AI to 25 or 50%.
Ex specials are initiated by pressing two of the attack buttons instead of one while initiating a special move. An example of this would be an EX hadouken being a quarter-circle toward the enemy and a light+medium punch.
Quick combos are executed by pressing in either the left or right stick. You should have default quick combos set up from the start. They take one bar each to use.
Naturally, you don't have to do these if you don't want to however I found this to be the best time to do them. If you've already completed them, just stick with initiating Pandora and getting the quick one hit wins.
It's worth noting that the no matter how you finish the opponent (with the exception of a perfect) it will be a Pandora finish.