300 Live matches will take a while but at least it's not as bad as the first Street Fighter IV's 500 match achievement :-)
If you haven't got anyone to boost with, you're probably quickest playing ranked matches via the Arcade mode's Fight Request system. Currently, there are still plenty of people online and the request system should serve you with a steady stream of competitors. To make it go quicker, you could consider playing 1 round matches although this is considered unsporting by some and there's a risk of getting hate mail. Very occasionally, you might find the request system gets stuck and doesn't match you up with anyone - if you play a couple of rounds against the computer and don't get the "A New Challenger..." message, exit to the main menu and start a new arcade session.
If you have someone to boost with, just set up an endless lobby with one round matches and this will go by very quicky. You don't need to reset the lobby each time, a new match is counted each time your characters are picked.
To keep track of your progress, select Player Data from the main menu and move to the Xbox Live tab. Don't confuse this with the Xbox Live Ranked tab - these matches need not be ranked or wins to count.