"Portraits" and "a portrait lens" are both very broad terms. Hang in with me for a few minutes and I'll explain.
Perspective management in portraiture is very important in order to avoid distortion that results in unflattering elongation, foreshortening, or misshaping of facial features or body proportions. Perspective is mainly control by workg distance between the camera (lens) and the subject. Selecting the proper focal length is important because it allows you to work at the proper distance in order to maintain a good perspective and negate distortion.
In any given composition, if you are work in too close you can, for example, elongate a subjects nose, exaggerate the size of hands or limbs of the are included in the composition or make a subject look unrealistically short or disproportionate in a full-length portrait. If you are too far away from you can compress the shape of the face so it looks more like a disk than a sphere- this lessens modelubg. Work too far away with very long lenses can minimize the illusion of depth because it tens to compress the distance between the subject and the background.
As for "portraits" that can be tight headshots, traditional head and shoulders (bust) composition, 3/4 length portraits, full-lenght portraits or group portraits with two or more subjects- perhaps family group images in full length. There are also case where you may want to include negative space and not fill the frame with the subject.
For your DX camera- good working distances can be established with these approximate focal lenghts as follows:
Tigh headshots and head and shoulders- 85mm to 105mm, 3/4 poses - 50mm to 75mm, full-lengh and groups 50mm. in extremely tight spaces where distance is limited by space, you can get away with 35mm for large groups but you need to very mindful of levelling the camera and maintaining vertical parallelity, which means not shoot tilting upward from very low positions or tilted down from very high positions.
Keeping someof these suggestions in mind, you may want to consider a zoom lens that will encompass the aforementioned focal lengths.
Another consideration is the lens's speed- maximum aperture. A fairly fast lens is desirable for selective focus control in that you can work a wide aperture to blur backgrounds, obtained "bokeh" effects. F.2.8 or thereabouts should suffice.
I fully understand that you need to consider the budget. I have seen many decent zooms in the new and used markets. I am very impressed with someof the Sigma "Art lenses but unfortunately, they are pricey- about $1400- new for a really cool zoom that would encompass all of the above focal lengths.
If there is only one fixed focal length lens in the budget- I would suggest the 85mm and work around the limitations.
Personally, although back inthe film days I did most of my portrait work on large and medium format gear my all-time favourite portrait lens for 35mm film is the 105mm f/2.8 Nikor. Loog ago when the idea of converting the studio to digital was starting to emerge, I picked up a simple Nikon D-300 with a kit zoom, just to get the feel of shooting digital. I plugged in my weatherbeaten, completely abused, cosmetically destroyed OLD 105mm leftover from my press days and used it to make a few shots on all my closeup headshots and head and shoulder sessions. I was using that lens for a paperweight and considered demoing it to a doorstop- was I ever surprised. I'll post a shot. I still have that old wreck and purchase a full-frame body just so I can use it. It's got a "look" that I like!
So, here is my very first DIGITAL portrait. My favourite and always-willing model and granddaughter was dressing and making up for her senior high school play and hammed it up for grandpa playing with his new toy and old lens. Time flies- she is now a registered therapist working in a veterans hospital and I am an old guy with a gray beard- still working in a studio!