Upgrading Legendary and Exotic weapons to maximum require you to earn Ascendant Energy (or Shards for armor), as well as one of the four materials in the game: Spin Metal, Helium Filaments, Spirit Bloom, and Relic Iron.
Ascendant Energy: You can find this in multiple ways, such as Gold Ratings on Public Events (check destinypublicevents.com), running Raid Missions, or doing Daily or Weekly missions at the highest setting. You can also breakdown Legendary or Exotic weapons to get this material, allowing you to indirectly convert your Marks into Energy.
Materials: These can be farmed on any given planet by going into Monster Closets (the caves and rooms in which enemies spawn) and opening any chests you may find. You can create a loop and just continue going around in a circle as the chests will quickly despawn and spawn somewhere else. Because of the speed at which the chests despawn, it's important that you get in sync with any other players doing the same, or you may end up in their wake and not get anything. Recommended Locations: Rocket Yards (Earth), Archer's Line (Moon), Ember Caves (Venus), and Scablands (Mars).
Once you have the appropriate materials, it's just a matter of leveling up. At a normal rate, without the Nightfall experience boost or without using a "Telemetry item" sold by Xur, you need (to my best knowledge) 212,500 experience to get an Exotic to full experience. This can take quite some time. Do Bounties, patrol missions, and daily missions to get there faster.
It's going to take a while, but it's generally worth it. Once you upgrade the last part of the weapon, you'll unlock this trophy.
NOTE: With the upcoming Dark Below expansion, Bungie has said that your Exotics will continue to be useful. This may mean that after the expansion, they may have further room for improvement, making this trophy take longer. This is NOT confirmed, but you have been warned that this may be a possibility.