The First Descendant

This achievement has a pre-req of Mastery Rank 3, which unlocks this option from the weapon workbench. To level a weapon up you'll need a Phase Exchanger. Talk to Anais on the right side of Albion from the spawn point and choose "Research Request" then "Enhancement Material" to find the Phase Exchangers. You can make a basic one to upgrade a Blue weapon or you'll need one of the higher rank ones for Purple weapons. Wait 2.5 minutes for the research to complete and collect it, then go to the workbench and select "Weapon Level Transmission" to begin. Choose any weapon, you'll see on that screen what materials are needed to level it up once, which should just be the Phase Exchanger we just researched and another Blue weapon of a higher level to consume. If you want to level it up multiple times, you'll need additional Phase Exchangers, but the achievement just requires a single level, so do that and on to the next!