I'd start in June or July if east bound and maybe early May if west bound. Those dates are probably the best shot at avoiding unpleasant weather and snowed in passes.

I'd also advise just riding alone, unless there is someone that you really want to ride with. You will meet folks along the way on the TA and will have opportunities to make friends and camp or ride with them. If you do arrange to ride with a stranger, I'd advise setting the expectation that you might split. I met a lot of folks that started out with someone and it didn't work out.