The Medic can seem like a tedious perk to level. Well no, in fact since the update for objective mode. It is now possible to get levels 1-6 within hours. Quite literally....
1. Set up a game of Steamland - Objective Mode (Normal or Hard)
2. Play the map until you are required to heal the Ringmaster
3. Continue to keep him alive and guide him to the extraction point.
As soon as he is at the extraction point, it is essential to keep 3 Clots or 2 Gorefasts alive. This is dependant on your current level. If you are level 1-3, keep 2 Clots or 1 Gorefast alive. Make sure the Ringmaster has the attention of these ZEDS. Continue to spam your syringe heal ability on him. Levelling from 1-5 should take roughly an hour if executed correctly. From 5-6 it's estimated around 3-4 hours. Throughout this process feel free to be in windowed mode and watch a film, or whatever floats your boat. This is quite literally the best way to level this particular perk, take advantage of it while you can.

Although, Please don't join a HOE server and call yourself a proffessional after this process. At least know how to be a level 6 medic before calling yourself one. Hope the guide helped.