Over simplifying it you have two main choices to work full time in deer management:
In England - working as a Forestry England Wildlife Ranger - do your DSC1,2 and the DMC or, ideally a course like one of these:
Game and Wildlife Management Archives
In Scotland - as above for DMQ quals then work as a seasonal ghillie (see jobs section on here like:
Vacancy: - Seasonal Ghillie, Braemore, Caithness) then look for roles as an understalker on private estates/with NGO to get your training. Otherwise above similar college route for Forestry & Land Scotland WR roles.
All will give you the experience and training you need if you then wanted to set up on your own as a contractor etc