You upgrade your exo suit in Single Player by earning points from completing challenges.
First off reloading checkpoints does not save progressThese challenges are very easy to obtain and are as follows:
Kill 1600 enemies
Get 100 Headshots
Kill 320 enemies with grenades
Acquire all intel.
I'd focus first on getting headshots as I managed to max them out not even half way through the campaign. Try the mission 'Manhunt' if you're having difficulty as this has both overdrive and lots of enemies.
After completing the story I had about 1400 kills, so you'll probably also like me have to re-do missions in order to get kills (If this is all you need, play mission 'Terminus' as you'll get many kills with the AST Mech.
All I can suggest about grenades is to throw 'em when you got 'em. The seond mission 'Atlas' allows you to re-run the training sim until you get your kills done and deosn't require checkpoint restarts.
If you need help finding the Intel, check this solution:
If you need help getting these requirements easily/quickly then please check out my video guide below!
Hope this helps guys!