I'll try to help out a bit here. I play almost exclusively that technique, and find it extremely versatile for many applications.
I'll try to get to the salient points of this playing style.
First, pick a song that utilizes this technique over an array of rythm and leads so as to get the general idea of how each is applied.
The one song that I highly recommend, which is like a textbook of this techqnue is Alan Jackson's "Who's Cheatin' Who". This is Brent Mason playing, what is what I consider the quintessential Chiken Pickin' showcase. It's also very do-able for beginners to pick up.
Make sure that you are fully engaging your pick, middle, and ring fingers. (Hybrid Picking) To get that really snappy tone, you've really got to use those fingers strongly, and really snap the strings. I also use acrylic glue on nails, since due to strong attack , it will eat my natural nails for breakfast in no time. So much of that chicken pickin' sound comes from the "snapping" nature of the picking fingers. Don't be afraid to almost over-do it, and then back off a bit to taste to find the sweet spot.
If you have been a flat picker up until now, it's going to take some time to "train" out of that mode. Take the time to slow down and re-learn how to do the licks to really take advantage of the extra fingers in play. Once you've got that down, it will eventually become second nature. For example, think about a picking run that goes from one string to the next. In the typical flat-picking style, one would simply pick all the notes with the pick. In hybrid picking, you may, for example, pick the first note of the next adjacent string with your middle finger. It's incredibly efficient when you nail it.
Other than that, I honestly feel the best way to approach this is to pick a song like "Who's Cheatin' Who" and just learn it bang-on. I think you will zero right in on it. I can't emphasize enough how great of a learning tool that one particular song is. I use that one to start off my guitar students also.
If you see the clips in my signature, they are 100% hybrid picked. It's about all I ever use now, since it's very efficient and opens up so many picking/rythmic possibilities.
Let me know if you have any more questions, I'd be happy to help.