thats an excellent question (and i don't think it would be possible to come up with DC's for the same manoever attempted on only wall... i remember doing that too.... it was always painful).
speaking from a bit of experience ( i jump a lot, including attempts between walls like that) i would definately have to say there are some strength checks that need to be made each time he pushes off the wall on the next one, because only your first initial jump is in fact a jump. The rest of your actions are feats of strength and i guess a little dexterity (balance maybe) because your turning every single time you move to another wall.
I would make it a strength check + half your jump modifier vs DC for a 10ft jump horizontally + a high jump of 5 feet up (maybe 3ft up per jump, it depends on your DM) + check attempt number (how many jumps youve done, it gets harder with each additional jump)
edit: in the numbers i changed it to a 5 foot horizontal jump because you are occupying 5 feet of the 10 ft space)
then make a climb check to hold the opposing wall good enough to brace for another jump (done it less then a second total)
likely between 15-25 for most straight walls (+5 if slippery)
if you fail your climb by 4 or less you are stuck holding the wall and cannot continue the jumping manoever, if you fail by 5 or more you fall.
Also, this would only be able to be attempted by a character carrying a light load... i cannot see anyone that is carryign enough to hamper movement somewhat being able to jump from wall to wall.
so a nice straight wall with lots of ample foot and hand holds would make the climb check a 15
and jumping in between should go something like this. (you could even thrown in a reflex save if you wanted, it would be appropriate)
jump check to jump up 5 feet in the air (with 20 ft running start) DC20
str + 1/2 jump check to shove yourself to opposing wall, 5 ft horiz plus 5 ft vert. DC 25
climb check to catch opposing wall. DC 15 (its a very nice wall

)
str + 1/2 jump check to shove yourself to opposing wall, 5 ft horiz plus 5 ft vert. DC 26
at this point you would be 15ft above the ground with your hands roughly 20ft above the ground (you would not have the full 8 ft reach as the jumping recoils a certain amount of you to be coiled to get the strength to jump again.
And it should have taken roughly 40/45 points of movement. You would have to take this as a full round action and have both hands free (so double your movement points). Armor checks should be applied to every roll and if you take any damage throughout the course of the jump you must make a concentration check as well as a climb check to avoid falling. You cannot continue jumping after taking damage, and you must either climb up or start again.
etc etc... this is being generous in the regards to assuming you don't need the 20 foot running start because of the momentum of your initial run and push (which you lose if you don't catch the wall perfectly.)
so in the long run.... its really really hard... you would be a lot better off simply climbing up.