I have a JET lathe that I purchased about a month ago. I was surprised that the amount of runout on the Inspection Sheet that came with the lathe showed the runout being .0004. I had the opportunity to check it out on a piece I made the other day. I had a 6" dia. piece of 7075 that was almost to the thickness that was called for on the print. I could only turn half the length back, then had to turn it around in the lathe and turn the remaining half. When you have runout, it is always real noticeable when you do this. I could not tell where the line was where I stopped on the first side. I'd be willing to say that the runout was next to nothing. For a Chinese made lathe, I was truly amazed.
With that being said, depending on your chuck, you should have the face of the chuck bolted to the body. You can loosen the bolts on the face and move it around very slightly to remove the runout. What you need to do to start is make sure everything is absolutely spotless when you do this. Not knowing how your chuck mounts, whether with three or four pins, or with a large spindle nut, if you have pins, make sure they are tight in your chuck. If a large spindle nut, make sure the spindle is free of any and all burrs. You can do this with a fine stone or a piece of fine sandpaper on a wooden block. Or remove the key turn the spindle on and polish it with some real fine sandpaper. What you will need now is a known large dowel or rod. I keep a guide pin out of a die shoe for this purpose. Mount your chuck to the lathe after everything is spotless. Chuck up the large pin or dowel and set up a tenth indicator on your compound. Loosen the face bolts on the chuck then snug them enough that the face won't move on it's own, but loose enough that you can tap it around with a plastic hammer. Once you get it to where you want it, tighten each face screw a little at a time until you have all securely tightened. To check the work, take a piece of round stock that is rigid. Use something around 6" long and around 1" in diameter. Turn about 3" of the stock taking off just enough to clean it up plus a few more thou. Shut the lathe off and move the compound towards the tailstock without changing the setting on the cross-slide. Turn your stock around then turn this end down. Both cuts should come together with no steps in the stock.
Checking the outside of the chuck for runout is not a good way to check a lathe for runout. One reason is the chuck components are all made in separate stages then bolted together. Doing that on different machines you end up with errors. Put all of those errors together and you will get something other than desired. If the chuck on your lathe was ever dropped, the lathe wrecked, or God forbid, someone leaving the chuck key in the chuck when they turned it on, will knock a chuck out of tolerance. Every once in a while, when I was working, I would check a chuck for runout and probably once a year I would have to refurbish it. I would tear it completely down, wash all of the parts off in the clean tank, stone off any burrs, use a good grease on the screw jaw plate and the tightening screws, reassemble and calibrate it. The bad is when multiple people run a machine and not pay attention. But for home, a good tune-up on your chuck will last a long time. If you do a lot of work on it that has interrupted cuts, you may want to check the chuck every few months. Other than that, a thorough tuneup once every year or two should suffice.
