What I perceive this to look like:
On the inventory screen, a section at the bottom titled say: Character Abilities.
Then a brief description of a set of abilities outlining each one such as follows:
Stamina - You have very little stamina and cannot jog for very long or carry very much.
Medical - You excel at medical procedures
Stealth - You have mastered the ability to creep without barely making a sound.
Hygiene - You have average abilities in maintaining cleanliness of equipment and tends to use more disinfectant than required.
Mechanical - You struggle with anything mechanical
How do you increase your abilities?
By surviving and carrying out actions on yourself, other players and items in the environment. The more you do these tasks, the better you become.
I have to caveat that with this:
Its takes A LONG TIME to progress, possibly months or even say a year to reach the top tier of skill/ability. I would hate to see character progression as something that could be maxed out in a week or a month.
How this would change the experience:
Squads could specialise into focused individuals, such as medic, mechanic, pilot, CQC weapons, sniper, thief, etc etc.
The lone wolf would be able to do the same and be able to negotiate services to others as a trade for items etc.
Your character would have value above and beyond just the gear in your backpack.
With the information on your abilities in your inventory screen it would make joining squads or squad recruitment outside the game easier and more viable as you would essentially have a CV you could give to prospective players to show your worth.
As a person currently in DayZ, you are worthless. Your gear has all the value. This needs to change, the people need to hold the most value over gear and items