There are eight level of renown:
Level 0 - Unknown
Level 1 - Unsung
Level 2 - Recognised
Level 3 - Familiar
Level 4 - Well-Known
Level 5 - Famous
Level 6 - Very Famous
Level 7 - Legendary
Your current Renown can be found by pressing Back > LB > LB > RT.
Renown is increased by completing quests, killing enemies, winning competitions, showing off Trophies and donating gold the the Temple of Avo.
Donating to the Temple of Avo only got me 100 renown for 100,000 gold. This is less than most side quests meaning buying the achievement is not an option.
I reached Legendary after completing the side quest Lost Trader just before the main story quest Rescue the Archaeologist. This is about three quarters of the way through the game. However, even if you only do the main story missions you will earn enough renown by the end of the game.