As
@Sharky says, chances are that you need to replace the cassette also.
If your cassette was worn when you replaced the chain, it will have done so in combination with the chain so the two will mate together quite well, however when you put a new chain on the cassette won't match well and the chain will skate over the top of it.
This can be quite dangerous as you may find yourself pitching forward suddenly if out of the saddle for example and also runs the risk of snapping your derailleur hanger and pitching the lot into the spokes on your wheel - this has happened to me, it's not fun.
Also possible, but less likely is that the chainring at the front will be worn, if that happens then the chain will do the same at the front. My suggestion would be to replace the cassette you are running now, and that should hopefully fix the problem, if it doesn't you haven't wasted any money as you will need the cassette sooner or later anyway.