Any play that you can feel is too much play. If you can actually see the bearing move, it's *way* too much play.
The axle is threaded. Loosen the clamp plate, then unthread the axle and pull it out. When you put it back on, make sure you tighten the upper nuts on the clamp plate first, then the lower. Best to use a torque wrench on the clamp nuts since over tightening will snap the studs.
While you're at it check, the rear wheel for play too. Grab the tire at the top and pull it back and forth in a side to side direction. If there's any play you should feel it.
If this is a new-to-you bike, plan on disassembling and regreasing the swingarm and linkage bearings soon.