No 'AF' lens will autofocus with that camera, you need 'AF-S' or a third party lens indicating that it has a motor in the lens.
Manually focusing on a DX camera is a pain. You can certainly do it, but the focusing screens aren't really designed for it and the viewfinders aren't particularly large. I'd not recommend spending good money towards one unless you also upgrade (or plan to shortly) to a body that supports autofocus on the lens.
Its not that you can't, its that its a pain.
If you're looking for a budget friendly upgrade from your kit lenses - try a Nikon 50mm f/1.8 or Nikon 50mm f/1.4. The 85mm is often used as a portrait lens and the 50mm on a DX body serves this purpose well. It won't be quite as tight for portraits, but you can get some stunning results. Plus, if its your only 'good' lens, its a bit more useful for general purposes*. I wouldn't worry about it overlapping with your kit's focal length - these lenses are just optically so much better that you won't care too much.
Additionally, you can go the other direction and consider something like the Tamron Macro 90mm f/2.8, if the focal length is important to you. You get nearly the same field of view as the 85mm and the ability to do macro shots (super duper close up shots of flowers, insects, water drops, etc). BUT, its a 'slower' lens at f/2.8. You won't be able to shoot quite as fast as the f/1.4 or f/1.8 lenses and you won't be able to get quite as 'creamy' a background.
Obviously, if you've got the money (2k US), you can consider the Nikon AF-S 85mm f/1.4 - it will autofocus. You can also consider the Sigma 85mm f/1.4 which will autofocus with your camera and is about half the cost of the Nikon at around 1k US.
All of these options are considerably better than the kit lens and provide autofocus options. Unless the AF 85mm f/1.8 is exactly the lens you want for a particular reason, I'd consider one of the above as a friendlier option.
*This isn't another 'everybody should have a 50mm' thing. Its just a budget friendly option that's an alternative to the suggested lenses.