There are as many ways to align the rear wheel as there are mechanics who do it. Here's mine, perfected over 60 years of riding, and a decade of racing. Once proficient it usually takes only five minutes, and does not require any special tools.
Preliminary alignment:
1: Set the slack as you want it, concentrating on the chain side primarily. Just try to set the brake side close, but don't get obsessive about it.
2: On a center stand of any sort (OEM, Phobstand, swingarm stand, a floor jack on the left with a block of wood under the side stand, or hang it from the rafters. It doesn't make a difference using this procedure as long as the rear tire is off the ground and the bike is stable and as level as possible.)
All adjustments in the following procedures will not significantly affect the chain slack you have already set as only the right adjuster is moved.
3: Spin the rear wheel backwards to keep the upper run of the chain tensioned and listen to the chain as it runs over the rear sprocket. If it is noisy slow the spin and look at the centering of the chain on the sprocket. If it's contacting the rearmost teeth of the sprocket on the inside the wheel is angled to the left. To adjust turn the right side adjuster to move the axle rearward a small amount, and spin the wheel until the chain gets quieter.
Conversely, if it is contacting the rearmost teeth on the outside of the sprocket the wheel is angled to the right. To adjust turn the right adjuster to move the axle forward too far, then while holding on to the frame kick the wheel forward (Do this carefully! Don't want to knock it off the stand!) to seat it against the adjuster. Now turn the right adjuster to bring the axle back, again spinning the wheel occasionally until it gets quiet and looks centered on the sprocket. Keep at it, invariably needing a kick forward if you overshoot, until it is as quiet as you can get it. Frankly, sound trumps centered. I have had a chain that was visibly centered on the sprocket, but was noisier than when it was biased to the inside. Left it there. I think it was a slightly out of line top guide.
4: Tighten the axle enough to cinch everything down, start the engine and run up the wheel while dragging the rear brake to stabilize the chain. It should make about the same noise level as it did while spinning by hand, just a higher pitch. If not, have someone take a short video of it that you can look at in order to determine which way it needs to go.
Torque the axle, clean up, and take a test ride. Done.
Rob
A P.S. for those amongst us who occasionally push the safety limits. I witnessed an incident where a mechanic decided that adjusting while the wheel is under power by using a long extension that would keep his hands away from the wheel is a better idea, and it worked for a short time until in a split second of inattention he let the socket come off the adjuster whereupon the extension pivoted into the wheel, caught a spoke, and was thrown into his right side chest. Fortunately he was using a 3/8ths breaker handle to turn it which prevented the extension from penetrating his chest, but it broke four ribs and not only punctured his lung, but actually lacerated it. It was touch and go for a day in the ER, but he did recover in only five months.
'nuff said.