Play hotzone medic. It has the lowest skill floor, the cheapest kit (under $2,000 [BETA]) and the most reliable payouts. Revives paid $400 [BETA] in the open, $800 [BETA] in the control zone and $1,600 [BETA] in the hotzone.
Take the free Emergency Resuscitator every life, carry bandages, and work behind cover near teammates. Bandage others, never yourself: self-aid pays nothing. At Medic 2, unlock the Field Resuscitator ($10,000 [S1] blueprint, $500 [S1] per use).
Second choice: when any faction reaches 20 points, drive your team's free spawn APC into the control zone and guard it. It paid about $3,000 [BETA] on arrival plus $100-$250 [BETA] per teammate who spawns on it, and you risk nothing of your own.
The payout figures come from the beta. The Season 1 changelog lists price and XP changes but no cash payout changes, and nobody has re-measured them since.