When something "goes to hell" it means it's deteriorating rapidly, or on a path headed for destruction. Example:
First it rained, then a raccoon got into the food; it looked like our whole picnic was going to hell.
"Going to hell in a handbag" (sometimes said as "going to hell in a handbasket") simply means going to hell a lot faster and without any effort! (Since a handbag is easy to carry.) Example:
Ever since Pee Wee Herman was caught watching that X-rated movie, his career has gone to hell in a handbag.
- For Teachers