This can be done in co-op or single player.
Here are some things that you should know first.
Completing the game in co-op will net you this trophy, but it WILL NOT unlock Master Mode as that requires you to play through on the Veteran difficulty solo.
Since the game doesn't take too long to finish, I suggest finding a co-op partner and doing a veteran co-op run together just to get used to the difficulty. Veteran wasn't hard, but it did seem like there were more enemies and everything was a tad bit more aggressive than they were on experienced difficulty.
TIPS FOR VETERAN
1. Use the Nemesis ship as those homing missles will easily be your best friend.
2. The boost will be your best friend. Remember to boost out of tight spots and save your bombs for the later levels (level 5 in particular).
3. If you can, save your overdrive shot for the bosses. While the overdrive shot doesn't do a lot of damage to the bosses on veteran, it can be used to cut through firepower (especially useful on the final boss in his third form).
4. Save as many humans as possible. Saving the humans will net you shields, extra lives, and bombs. Trust me, you will make a dumb mistake or two and you'll be thankful that you have a few extra lives in your pocket.
5. Use bombs wisely (Star Fox reference). I mentioned in TIP number 2 that you should save your bombs for the later levels, but if things REALLY get hairy, then fire a bomb off. It's better to waste a bomb instead of wasting a life. Remember, saving humans will net you 1 bomb and 1 extra life per level.