This is a pretty tough achievement to get so be prepared for the long haul. This achievement requires you to upgrade all weapons three times each. The weapons are: Garand, Thompson, Gewehr, Springfield, BAR, Karabiner 98K, Stg 44, M12 Shotgun, MP40, Colt .45, and C96 Mauser. You also have to upgrade the three types of grenades (Gammon, Frag, and Stick) as well as the Panzershrek and Recoilless Rifle.
My suggestion to start is to just tear through the game on whatever difficulty you want, but Casual is probably the way to go for most. By the time you beat the game you will probably have fully upgraded about 5 weapons. Go back to Operation Husky (the first level) and play it over and over again to fill up the rest of the ones you can. Unfortunately a few weapons are only in certain places and these are the toughest to upgrade. Remember that you HAVE to reach a checkpoint for the upgrade to keep valid if you die.
1. Recoilless Rifle: Located only in Operation Neptune. Taking out the tank helps big time.
2. Panzershrek: Located in Operation Market Garden (on the bridge) and Der Flakturm. Try to get kills in groups.
3. C96 Mauser: Very tough to upgrade because it is found in Operation Varsity in a safe. You can only use it in Varsity and Flakturm. The gun isn't very powerful and enemies take quite a few shots to take down. This can be very tedious. Try hard for headshots or melee kills. I found that Der Flakturm works best because the enemies tend to group up.
I also found the Frag grenades tough to upgrade because you tend not to get extra after running out. Try to get groups as individual kills barely make the bar go up. I found Operation Neptune the best for these because you get quite a few and the enemies are in enclosed bunkers and can be easier to kill with grenades.
*Please see comments for additional tips provided by commenters*