For this trophy you need to win by an innings, which means enforcing something called a follow-on.
In order to enforce this, you need to set a sizable total batting first, and ensure that you bowl the other team out for at least 200 less than your first innings total. You will then be given the option to enforce the follow-on and make them bat again (the game will also give you a helpful indication of how many runs the other team needs to avoid the follow on too for a point of reference). Bowl the team out again before they reach your first innings total on aggregate (e.g. you make 500 and they make 100 and 399 in their two attempts at bat), and you will win by an innings.
This can be done during the Ashes and will probably occur naturally on Easy difficulty (which you should be playing on if you're looking to get this game over with as quickly as possible), but be warned. On the 360 version I enforced the follow-on in one of the matches, ended up winning the series 5-0 and didn't get the achievement for winning the Ashes at the end, and I soon found after trawling through gamefaqs that this can happen if you enforce the follow-on, so I had to repeat the entire series again without enforcing it.
I'm not 100% sure that it voids Ashes Champion on the PS3 version (or even 100% of the time on 360 for that matter), but just to be safe I wouldn't enforce it during that series; rather, play a separate Test match and do it there.