Europeans with six weeks total annual vacation and short-hop budget airfares can afford to do all-hiking vacations, but for Americans, especially if you only have a week or ten days for the trip, I personally would be inclined to do some hiking and a good deal more sightseeing, to make it worth the flight over. Maybe the wall a day or two, part on foot and part by bus, but the rest of the time at other stops in the border area. (Unless you were planning a total of two or three weeks, in which case a week's hike would make more sense.) York, Whitby, Lindisfarne, Durham, some of the Scottish border towns with keeps and so on, and Edinburgh and maybe even Glasgow -- if there's any of that that you haven't been to, you might want to consider it all before deciding to commit a whole week to Hadrian. It would still fit with the theme, and you could go back another year for things farther north in Scotland (highlands and islands, including Orkney -- all well worth a separate trip), or farther south in England.
Of course, if you've already seen most of those places and are looking for somewhere different, disregard this post and go hike. (-:
The other thing to keep in mind, besides views of sheep, might be the weather -- it would be worth checking what kind of equipment is recommended for the time of year you plan to go. (My guess: Plenty of rainwear. (-; )