To level a subclass, You want to start the Game and create a Character out of the three available (Hunter, Titan and Warlock).
You can have three Characters at the same Time and if You want to, You may have three of a different Class or even three of the same Class, but anyway!
Each Class offers two subclasses:
Titan with Defender and Striker
Warlock with Sunsinger and Voidwalker and
Hunter with Gunslinger and Bladedancer!
Those subclasses could be switched at any time You like, but i'd prefer to level them up one after another.
Every Subclass has different Perks, You level them up, by gaining XP from various Events like turning in Bounties, doing Raids and Strikes, playing MP and Story Missions.
The Moment you've unlocked the last Perk from a specific Subclass is the Moment You earn that Trophy!
it shouldn't take too long, but be prepared to put in some hours - like everything else in Destiny it requires some grinding