Hi Tam - it really depends... the official TAT route as it stands now (from West Virginia to Oregon) is going to take you longer than four weeks realistically - certainly if you've any hope of actually seeing/absorbing anything, and not riding 16 hours a day...
Of course this time of year you are going to have to miss out large chunks of it - and typically those that coincide with the slower more technical terrain (in Colorado, and higher elevation in Utah and Idaho), but even then, 30 days would be pushing it from coast to coast.
If you want an illustration, there is my
ride report from 2015 here on the HUBB - it took me 25 days on the actual TAT itself, but that was starting at the previous official start in Andrews NC (there has been another 500+ miles of TAT route added since then).
I'd say that was a realistic solo pace if you're confident in your riding skills and have packed light - oh, and aren't planning on camping much either... but even then I only had one day off (not riding), and towards the end was covering 300+ mile days some times just to make my target finish.
Ultimately though, if you have the maps and/or the GPS route, you can always streamline sections to stay on target - although I always say that's a shame since even the shortest section you might choose to miss could actually offer a highlight of your trip for one reason or another?
If you want my suggestion to cover as much TAT as possible earlier in the season, then I would ride from the east coast (the Blue Ridge Parkway is always scenic) and as much of the TAT route as you can in Virginia, North Carolina and Tennessee; and then streamline through Mississippi (this is basically what I did last spring on the G310GS -
ride report here).
Pick the TAT up again once you cross in Arkansas and include the first couple of sections north in Oklahoma too - then straight-line on highway 412 from Tulsa to Boise City - that should see you in New Mexico within two days.
Realistically you'll only get as far as Trinidad (or possibly La Veta) on the TAT route in Colorado before the altitude gain means you'll need to hit the Highway again - my suggestion would be to stay south on 160 for Durango (ride the train for half a day!), then pick up the TAT again once you cross into Utah as I suggested above...
That way, you ought to be close to Moab around the 2+ week mark, which means it's feasible to ride as much of the rest of the TAT as possible and make it to the coast within a month.
But don't bust your arse or the bike just to try and keep to a tight schedule - it would be far better to miss out sections of trail than blast through them and for it all to become a blur... and it's also a shame to not have at least a little time to stop at some of the trail/road-side points of interest, since it really is the whole point of following this particular route after all?
So in that regard, another example of a short-cut to keep you heading west - once you reach Promontory Summit in Utah (definitely worth the visit, and essential if you love steam trains!), you can always forfeit the trail sections north of there through Craters of the Moon and across the mountains towards Sun Valley ID, and instead take I84 towards Boise and pick up the TAT again in that region for the final run west through Oregon to the coast.
Hope that helps!
Jenny x