There are a couple ways to go about getting this achievement. You can use the Pilot Training option, found in the shop, and buy the Pilot Points necessary to level up your character. You'll need quite a bit of G to get there, though. The other way is to grind the experience yourself by doing missions over and over, and obviously you can mix and match and do a little of both.
First of all, skills you'll need and how to unlock them:
Little Giant (Initially available) - Gain a boost when fighting mobile armors
Pilot Sense (Beat Memorial Missions to unlock) - Gain more Pilot Points at the end of a mission
Ace Killer (Memorial Missions) - Gain a boost when fighting enemy pilots
One-man Army (Memorial Missions) - Gain a boost fighting large groups of enemies
Salvage Connoisseur (Judau level 5 Friendship) - Gain more G at the end of a mission
Final Sacrifice (Treize level 5 Friendship) - Large boost to combat ability, but cannot heal even with items
Option 1: All for the Money
Recommended skills:
Pilot Sense, Ace Killer, One-man Army, Salvage Connoisseur
For this, the Special Missions are your friend. There are three specific missions you'll need, but since they appear randomly, you'll be playing other missions too. The three missions you'll want to look out for are:
Supply and Demand (S&D)
The Rising Cost of Mobile Suits! (RCMS)
Junkyard Battle! (JB)
Again, all three of these appear randomly, but you'll have to clear the lower ones for the others to start unlocking. The plus side is that you only have to clear them on one character and they'll start appearing for others. In other words, beating S&D will have RCMS start appearing for all other characters, and likewise with JB.
The trick then is getting them to appear by completing other missions. I recommend beating the Collection Missions, because the low-level ones go by pretty quick. Properly equipped, you can go through Heroes of the One-Year War and Mourning Warriors in under five minutes. Equip Pilot Sense, Ace Killer, and One-man Army for the quickest times. If you prefer, beating the Trial of the Giants can also go pretty quickly (see below).
When going through the Special Missions, choose your best suit and equip the following skills: Pilot Sense, Salvage Connoisseur, Ace Killer/One-man Army. This will net you the most G to use in Training. In terms of G, S&D gives about 5,000-7,000; RCMS nets you anywhere between 10,000 and 15,000; and JB will give you 18,000-20,000, with twenty thousand being the cap for any single mission.
Option 2: Learn through Experience
Recommended skills:
Pilot Sense, Little Giant, Final Sacrifice
For this one, the Challenge Missions are the way to go. Specifically, you want to play Trial of the Giants 2 and 3. TotG 1 and 4 have you go up against the Big Zam and Dark Gundam mobile armors, and those have nasty one-hit KO moves when they're low on health. You only get one life each mission, so those two are a no-go.
Pick whichever pilot and suit you like best and go for either TotG 2 or 3. TotG 2 has you going against the Psycho Gundam Mark 1 and 2, and TotG 3 has you going against two Queen Mansas. I prefer going against the Psycho Gundams, but the Queen Mansas are also pretty easy.
With a level 4, tier-one mobile suit and the above skills, you'll beat each mission in under two minutes and earn anywhere between 4000 and 6000 Pilot Points each time. The description for Final Sacrifice is wrong, because you'll still recover lost health through natural healing (repairing?). And since the mobile armors don't drop items and you're only going against two of them, that drawback is a moot point.
And again, you can always go with some of Column A and some of Column B. Whichever floats you boat and all that.