Roundness problems are usually caused by running the part too low on the workblade and possibly by running the reg wheel so fast that the part is not given enough time to round up well. It can also be caused by improper wheel selection and other things.
On the other hand, the chatter suggests that you are too high on the blade. I'm sure you know the center height of the wheels - rule of thumb is that the part is no more than 1/2 the diameter above C/L.
If you have recently changed the grinding wheel, you may have selected a wheel that is a little too hard for the application. It is also possible that your reg wheel speed is too fast. Another potential culprit is that your coolant concentration is too low. Also try changing your dressing wheel and/or reg wheel diamonds. Dull diamonds do a poor job of resurfacing the wheels.
Also, be sure to check that the part is in the groove on the blade. If it is not in the groove (worn by many parts over time) you will have all sorts of problems. Check this by high-spotting a slug and, resting it against the reg wheel, see if it is in the center of the groove.
If you have recently changed the blade, check that it is parallel to the fixture on both entry and exit sides. A blade that is not level with respect to the fixture gives a wide number of problems. A bent blade is another possiblity. If the grinding wheel ever touches the blade, you will have to change blades as it will either bow or chip off the carbide.
Taper problems can also be caused by poor setting of the guides on the reg wheel side. A guide angled toward or away from the wheel will cause taper on one end.
Be sure the reg wheel is properly dressed. Everything else can be OK but a poorly dressed reg wheel will cause geometry problems.
Simplest machine in the world but there are many variables that must be correct in order to get good parts.
Steve