You can kill the same named enemy multiple times and you must be the person that gets the last hit on them.
Here's a image that roughly shows where the named enemies spawn.
The site the map is from also details the different named enemies:http://www.gosunoob.com/division/dark-zone-named-enemy-locat...
The map was more useful for me but I'll leave the link to the page.
Even with the map above the named enemies can wander around a bit and you need to factor in that other players can kill them, I think it takes about ten minutes for them to respawn.
If you're fighting one in the open street I suggest staying far away and picking off at them with a sniper. If you're fighting a subway named enemy try to lure them to the stairs, some times they'll just stand at the foot of the stairs but even if they don't you got nice cover and some time to stall as they should pause going half way up.
The lower the zone number the easier the named enemy will be, so stick to zones 01/02/03 if you're playing solo.
Upon a successful named enemy kill your achievement tracker should go up by 10%.
Took me about three or four hours running around randomly to kill the ten required.