For the uninitiated, the term Catskill fly doesn't refer to any one particular fly pattern. It is a style of tying flies that originated in the Catskill Mountain region of New York State over one hundred years ago. This area is considered the birth place of dry fly fishing in America when Theodore Gordon first copied & modified the designs of some British dry flies. These days, almost any mayfly can be tied in the Catskill style of dry fly.
There are numerous ways to tie this style of fly. Below is a video demonstration showing how a Hendrickson dry fly can be tied in the Catskill style. The methods shown here will work for any other Catskill dry fly pattern. So use these methods to tie up your Adams, Light Cahill, etc.
Note: if you are a beginner fly tier, I wouldn't recommend you try a dry fly as your first attempt to tie a fly. Find a good teacher, a good book, some videos, & start with an easier pattern like a wooly bugger.