This requires you to level up an individual character a total 7 times.
Once you hit level 60 go to Clea (The Cube Shard Vendor) in a hub and purchase the Ancient Gourd of Prestige for 10 Cube Shards. You can earn Cube shards by completing a any terminal for the first time each day on each difficulty.
When you use the gourd it resets your character back to level 1 with a new color for your name in this order:
Begin: White
1: Green
2: Blue
3: Purple
4: Orange
5: Red
6: Yellow (Cosmic)
You will earn XP at the normal rate across all prestige levels with the exception of Cosmic Prestige. Once you reset to level 1 Cosmic Prestige you will earn XP at 1/25 the rate thus taking you 25x longer to level your character to 60.
Make sure to save up XP boosts of any kind or work on getting other classes up to 60 first to get the passive XP gains before attempting this to speed up the process.
Once you hit 60 on Cosmic the achievement is yours.