Bike chain too long?
You probably need a new cassette/freewheel. A new chain on a worn cassette will jump (especially on the harder gears).

The harder gears have two things going against them. 
Less points of contact to share the load. 
Less points of contact so they tend to be worn the most.

If you have front derailleur then using the bigger ring there will let you use bigger cogs in the rear.  If I can be in 3 or higher in the rear on the big ring up front then I use the big ring up front to keep miles off the smaller cogs in back.