The Starrett 98 Series machinist level has a sensitivity of .005"/ft. Their 199 Series , .0005"/ft. The inspection grade Empire digital level claims .01º translates to .002"/ft. but whether this is truly achievable is questionable. Aside from sensitivity of the vial, the accuracy of the machined surface enters in. A carpenters level often has an extruded or, at best, milled surface.
A carpenter's level has about 10% of the sensitivity of the 98 Series level. A barely noticeable deflection of my carpenter's level puts my .0006"/ft. machinists level off scale.
Nonetheless, I would start by leveling the lathe. A lathe can have a twisted headstock and attempting to correct this by twisting the bed will result in a corkscrewed lathe. Truly, two wrongs don't make a right.
I would first level the lathe to the best of my ability. With care, even a carpenter's level should get you within .01"/ft. Next I would look at headstock alignment. I would do this by putting the faceplate on the spindle and sweeping across the surface. Alternatively, you can mount a carriage bolt to the faceplate at a suitable distance from the center. Rotate the faceplate so the bolt is to the front and horizontal with the spindle.
Mount a dial or rest indicator on the cross slide and zero out on the bolt head. Lock the carriage and recheck zero. Now, rotate the faceplate 180º so it is on the backside of the lathe and move the cross slide to intersect it. The reading on the indicator should be zero. (Note: there are some schools that maintain that a lathe should face a slightly concave surface, the reason being that such a surface won't rock when placed on a flat surface. I didn't go to that school) My G0602 has an adjustment for headstock alignment, others do not. I would first align my headstock for equal readings front and back.
There is the possibility that the cross slide ways aren't perpendicular to the bed ways. Checking that would require a known good square. Correcting a non square issue with the cross feed most likely would involve something like scraping the cross feed which is fairly advanced work.
Now, I would move on to the ways, checking for taper by either the two collar or the RDM method, and shimming as required. I would go back and check my faceplate measurements, readjust if necessary and recheck the ways.