You start the game with three Medkits. Which means that you have 97 left. They can be used at any time in a contract so long as you have at least 1 health point missing. You can get Medikits in two ways, buying, or gifts.
You can gift 5 Medikits per day to an Xbox Live friend.
Alternatively, MedKits can be purchased in bulk:
1 Medkit - $200
5 Medkits - $750
20 Medkits - $2500
50 Medkits - 12 Contract Credits*
*Contract Credits are the "Premium Currency" of the game. And can also be bought in bulk for Microsoft Points:
30 Credits - 160 Microsoft Points
90 Credits - 400 Microsoft Points
190 Credits - 800 Microsoft Points
400 Credits - 1600 Microsoft Points
1000 Credits - 3200 Microsoft Points
Therefore, to buy enough Medkits to use for this achievement would set you back:
1 Medkit at $200 x 97 = $19,400
5 Medkits at $750 x 20 = $15,000
20 Medkits at $2,500 x5 = $12,500
50 Medkits at 12 Credits x 2 = 24 Credits = <160 Microsoft Points
To see how many Medkits you have used, go to your safe house and select the stats tab along the bottom. From there, scroll all the way down and it's the last stat that's tracked.
A good way to work towards this is to start a contract that requires to eliminate a "Boss" AND his henchman after you have some Medkits ready, then Kill/Capture the boss which will gain the henchmen's attention, they will shoot and damage you, and you can heal by pressing the heal button just to the right of the scope button.