Categories are unlocked just by leveling up the weapon. Each of the 10 categories unlocks at different weapon levels.
The 10 categories:
Spray Paint
Woodland
Digital
Urban
Splinter
Topo
Tiger
Stripes
Reptile
Skulls
Once you unlock a category by reaching a certain weapon level, each of the categories then have 10 variants of camos to unlock. Each camo is tied to in-game challenges (e.g. spray paints require certain kills, others need headshots).
In total, that's 100 camos - you can unlock 3 more:
Gold camo - unlock all 100 camos for each weapon
Platinum camo - unlock gold for all weapons in a category
Damascus - unlock every camo for every weapon