From Gaming Asylum Wiki
Infamy is the rebel equivalent of honor or prestige, a special experience that you gain by doing illegal and rebel-based activities around the island.
There are 4 classes: Specialist, Medic, Drug Lord and Arms Dealer.
Must have a rebel license to learn and earn Infamy.
You can earn infamy by doing various activities like: capturing cartels, banks, processing, etc.
When earning Infamy, it will not always show up in chat for small things.