With a DC 15 check, you can increase your base speed by 5 feet for one round. You may normally only attempt this check once per minute, however, if you beat the DC by 5 or more, you may attempt to continue the sprint by making another check next round.
For each full 5 points by which you beat the DC, you may increase your speed by an additional 5 feet.
What do you think? It couldn't hurt to have another Strength-based skill and this one seems like it would be a good way to nix the need for the Run feat. Let's be honest, no one ever took the Run feat anyway, but I could see PCs investing a few skill points in Sprint from time to time. Barbarian, druid, fighter, monk, paladin, and ranger would receive it as a class skill. Rogues might be worthy as well.
I don't like the mechanic. I'd have the speed increase scale:
+0 (less than 10, fatigued 1 minute)
+5 ft (DC 10, fatigued 1 minute)
+5 ft (DC 15+1/successive rnd)
+10 ft (DC 20, fatigued for 1 minute)
+10 ft (DC 25+1/successive rnd)
This skill either cannot be used when fatigued, or the skill is penalized (-5?) if used when fatigued.