Hi Paul,
I am not sure how flexible your small lathe is, but there is a cardinal rule for all lathes, and that is that the spindle axis must be aligned with the bed, and hopefully, the tailstock point will also move along this line, wherever it is placed.
Commonly this alignment is said to require a precision level, levelling the bed at a location near the headstock and another location near the far end of the bed. You can, however do a more accurate job of setting up your lathe by using trial cuts on a sample shaft.
Bolting your benchtop lathe down to a counter, or any kind of base, can twist the bed out of true, and this changes the line of approach that the carriage makes as it advances towards the spindle.
This first test must be done with a piece of stock held only in a chuck. Use a fairly decent diameter stock, say 1" or larger, with 4" or so projecting from the chuck (we don't want it to chatter when cutting). Neck the projecting stock, leaving a 1/4" or so full dia on the very end, and also full dia for the last inch next to the chuck jaws. Now, sharpen up a real keen toolbit edge, with a very small tip radius (to reduce chattering tendencies). Take a very light cut on the outer end of the piece, enough to clean it up, and without touching the tool setting, wind the carriage acrross to the zone near the chuck and cut that diameter at the same tool setting. The surface finish needs to be very smooth, but do not polish or file it. Cutting with oil on the surface will help.
Measure the diameters of these freshly machined surfaces and compare the readings. (Please tell me you have micrometers! ) If you detect a difference in size, this indicates that your lathe is cutting a "natural taper" which you can eliminate by shimming the "feet" of your lathe bed, usually at the tailstock end.
If the outer end diameter of your piece is smaller than the diameter near the chuck, you can imagine that your carriage is approaching from a diagonal drawn from the back side of the bed at the tailstock end, across the bed to the front corner of the headstock. To correct this, shim the back leg of the tailstock end of the bed.
You need to take this advice with a grain of salt, as other factors can enter in, such as the keeness of your tool, how well the chuck jaws actually grip the surface (so the bar does not wobble while being cut).
After you think you have this set properly, take more trial cuts to prove it. Or, bore a deep hole in a piece of pipe, and measure the diameter front and back. You want this perfect, as it is a bearcat to straighten out taper bores.
After you have this working repeatably, then you can do the tailstock alignment procedure that most likely you know about. Turn a test piece between centers for this and adjust the tailstock's sideways adjustment screws (perhaps buried under heavy paint putty near the base of the tailstock).
Notice, I said turn between centers because a piece held between chuck and tailcenter is likely not very accurate, at least for testing. This is because any eccentricity of the test piece center hole will flex the tailcenter, because the chuck is holding it stiffly. This will create an unexplained (up until now ) taper on your part, even if your lathe is perfectly set up. You can check for this effect when machining day to day workpieces by placing a dial indicator on the tailstock spindle, with the tail center engaging the workpiece (held in the chuck) and rotating the part one revolution.
If there is any misalignment of the part, it will show up as flexing displayed by your dial indicator. For moderate workpiece misalignment, you can use a soft face hammer to move the part a bit in the chuck, until tailstock flexure is near zero. Then you should be set to go on your previously perfected lathe setup.