It should be possible to kayak from the sea to Croft on Tees or maybe further.
There are a few things in the way, like the Tees Barrage, and a weir about two miles downstream from Middleton One Row, but you can get out and walk round them easily enough. There needs to be a reasonable volume of water in the river because it's quite shallow in the summer around Middleton One Row. Further upstream from MOR at Sockburn and Fishlocks it's a bit narrower, and therefore deeper, and (although I don't kayak) I would have thought it would be easy going. After a couple of giant meanders, you go past Neasham, Hurworth and then Croft. I have seen people kayaking at Croft. That's the confluence of the Tees and Skerne. The Tees above that point is shallower, up to the pumping station at Broken Scar, just west of Darlington. Above that I don't know, although I have seen people kayaking up at Greta Bridge, which is a good distance further upstream.
Good luck if you do it. Just near Girsby is the most wonderful little suspension bridge - just really a footbridge, but beautifully designed. A bit further upstream from there is the road bridge at Over Dinsdale - replaced in the 1950s, but the original was fitted with explosives in 1940 in case of a German invasion. A German invasion - there? I wonder where they thought the Germans were coming from and going to.