+1 for ladders.
I have to do a fitness test every year at work, including chin ups (changed to pull ups a couple years ago) and dips (changed to push ups). The passing standards are pretty low, but years ago, I used to train specifically for the test to try and get my name up on the record board (the top three scores each year for every category were put on a plaque and hung on the wall. These plaques went back about 20 years.)
The most chin ups I ever did was 46 (strict, no kipping allowed) and dips was up around 60.
My training for the test was pretty simple. Starting about 4-6 weeks out, I would change my training to just the exercises used in the test and train them using a ladder, well a pyramid actually. For chins, I would do 1, rest for 10 seconds, do 2, rest 10 sec, do 3, rest, etc. Once I got to a number I didn't think I could beat, I would go back down. I also used several different grips, normal grip, close grip, behind the neck and commando pull ups.
For dips, I would do the same technique, but go up and down by 2s. Same for push ups, using normal, close grip, wide grip, and dive bomber.
Usually, the first week, I would do this maybe 3 times per week, but by the end, I was doing it 4-5 times per week. I would also do a set to failure to start and end the workout. Each day I would try to beat the previous failure set by at least a rep and, if possible, more.
This training method works extremely well and fast, and each year I would go from about 20 - 40+ chins in about 4-6 weeks. A serious word of caution, though, doing extremely high volume chins and dips like this wreaks havoc on shoulders and it would take me a couple weeks of no chins/dips and shoulder friendly movements to get them proper again - which is why I only did this for 4-6 weeks each year.