I can help you out here. I'll give you a little background on myself. Back in 2002 I was in the same boat as you. I can't remember how many dips I could do, but I could only do 10 chins as well. I followed the program I'll mention later and within a couple of weeks I was doing 20 chins. I have to do a fitness test for work each year and it includes chins and dips (pushups replaced dips last year though). I generally only worked chins and dips in the 5-10 rep range (weighted) except when preparing for the test. I spend 4-6 weeks doing the following program and in '04 I got 32 chins and 43 dips. My records are 46 chins in '07 and 60 dips in '08.
These are without kipping. I tried that once during a workout and thought I was going to injure myself, lol. I have terrible shoulder flexibility/mobility.
The thing I found with chins and dips, is that you get better at them by doing chins and dips. Even though my max numbers are high, it's not like that year round. Most of the year my max sits around 20 chins/30 dips.
Here's the program. All chin up exercises are done in a pyramid fashion with a 10 sec rest between sets ie. 1 rep, 10 sec rest, 2 reps, 10 sec rest, 3, 4, 5... Once you reach your max, you just go back down ...5, 4, 3, 2, 1.
Close grip pull ups
Behind the neck pull ups
Commandos (grip the bar sideways with your hands together. Pull up to a shoulder, down, up to the other).
Dips (Increase the steps to reps of 2 -- 2, 4, 6, 8, 10...8, 6, 4, 2)
Push ups (Steps of 2 as well)
You should be able to go from 10 - 20 pretty quickly with this. After 20, if you want to keep going you can, but one change I would make is starting and ending the chin and dip portion with a max set.
Make sure you're only taking a 10 sec rest between sets and the key here is to push yourself through the burn/pain to try for that extra rep.
This is a pretty intense workout and expect to be sore for a couple of days after your first couple of workouts. If you need to take 2 or 3 days off between the first couple of workouts, no problem. As your conditioning and recovery improves you can move down to 1 or 2 days and then 1 day off in between.
***NOTE: When I'm preparing for my testing, this is the only workout I do apart from running and sit ups, which are also tested. I only do it for the 4-6 weeks before the test and switch to a balanced routine for the rest of the year.
Only doing chins/dips/pushups can lead to some serious imbalances and I've had to nurse shoulder injuries in the past. Mind you, I'm pushing for high numbers. You should be able to get to 20 without worry.
Let me know if you have any questions and keep us updated on your progress.