As you can tell from varied responses, there are a bunch of things that can and will cause vibrations. If you are bottoming out you can be low on shock oil, your springs are shot, you need more air pressure, or your anti dives are not working. It is highly recommended that you rebuild your forks and install Progressive springs. Inspect all your parts to make sure you don't have a bent tube, etc. and make sure the new oil you put in is the same amount in both shocks. If the both front shocks are not equal in oil level AND air pressure it can cause handling issues as one side may act different than the other causing different recovery rates etc. from a bump; one side will rise different from the other side.
Proper tire balance is very important for minimizing front end oscillations as well as the tire itself! There have been cases of people installing a new tire then finding out the tire for some reason or another caused handling issues. Also never mix radials and bias tires as it can and will cause handling issues.
Yes, with that old of a bike I would replace the head bearings and races. You are talking 28 year old parts and just as old grease. Also make sure the torque on the head nut is at the proper spec. Too loose or too tight will cause oscillations. Member Dingy sells a really handy tool just for this purpose.
Another way to reduce oscillations is by stiffening up your fork brace by replacing it with either a Superbrace or one that member Condor makes. The one Condor makes is just as good as the Superbrace and a LOT cheaper!
Lastly, as mentioned, the rear suspension can effect handling as well. Make sure there is no sideplay with your swingarm, and that the mounting nuts haven't loosened. Replacing the swingarm bearings is a good idea. You will also want to check out all your bushings and sleeves on the rear shock mount assembly while you are at it. Many of us have drilled and tapped for grease fittings at each sleeve. The nylon sleeves usually don't go bad as much as the metal bushings developing wear. Parts are getting scarce though.
All these things add up to effect the overall handling of the bike...