If you are trying to increase the strength of your biceps for your "Pull-ups" (palms away on the bar) then I would suggest doing "Chin-ups" (palms toward you on the bar).
Train them first in your session. Do three sets of Chin-ups. Stop 2 reps before failure. Then do one set of Pull-ups. That's four total sets.
I don't like working with weight every session if you are trying to go for reps (as you stated).
Do the Chin Pull-up rotine three times per week. If you feel you really want to work with weight keep it light 10lbs. to 15lbs. is enough.
to hit the 25 reps (which is your goal) you need to get the body used to flushing out the lactic acid. And you also need to work the proper muslce fibers as well. These things do not get trained properly (for high reps) when you work with heavy weights and low reps.