To Unlock Ajax and his Character Mission in Black Ops 4 Blackout, you need to complete the Character Mission that is specifically attached to Ajax.
His task is relatively simple compared to others, Ajax required to find 3 Destroyed Armor Platings and finish the match with them in your inventory. These Armor Platings only drop from enemy players in Blackout. If you are lucky you can even scavenge more that one from an opponent, but it is also possible that there is nothing to grab.
Once you have at least 3 Destroyed Armor Platings on you, make sure you keep them in your inventory and let the match play out as usual.