To see how much damage you have dealt with each Legend:
1. From the main menu, select the "Legends" tab.
2. Choose the character that you would like to see.
3. At the top of the screen, select the "Banners" tab.
4. Scroll down and select "Tracker 1". This will show you all the trackers locked/unlocked for that character.
5. Scroll down in this list to see the "Damage Done" tracker.
For this achievement, you must have 5000 damage dealt for each of the 8 characters in Apex: Legends (including the two locked characters).
NOTE: Damage to a player's shield does not count toward the "Damage Done" stat. Only damage done to the player's health progresses this stat.
NOTE 2: @Shadow2327 has confirmed in the comments that the new character, Octane, counts towards this achievement. This means that you only need 5,000 damage with your choice of 8 of the 9 total legends.