So for the first part of going for these trophies, I was just 'playing the game'. I then discovered there is a very quick way to get these and managed to go from rank 0 to rank 2 in 30 minutes (One match)!
This was much faster than the 2-3 hours it took me before knowing this.
Double XP was on when I did it which seems to be on every Saturday, though I'm not sure whether double XP only affects online rank or also class rank.
On to the Steps;
1) Go to the "Soldier" tab on the main menu, then go into "Service assignments"
2) At the top there is a "Getting Started" list that has a 50k XP reward for each of the 4 infantry classes. You can press square to track whichever one you like.
3) Focus solely on ticking off the checklist for that assignment. When completed, the 50k XP will count towards your class rank. I got both of my level ups at once as soon as this assignment completed so that is all that is required.
4) Even if my double XP did affect this, even at normal XP this should be enough to get 1 rank up, so you will have this trophy by the time you do it with all four classes.