How to level up the field medic perk without a team.
Steamland Objective Mode
You can level up field medic by playing the map Steamland on objective mode by healing the Ringmaster. I found that the best way to do this task is to leave a few Clots alive attacking him and sit back with a medic gun and keep healing him. If You're playing as field medic use your genades while moving through the map to heal yourself and never use the syringe on yourself (the grenades heal you but will count as healing another player).