I have bought a KMC Z8S chain , but it slides over the rear cassette . The model of the cassette is Shimano CS-HG30-81 , it is not worn yet , and the chain length is also perfectly calculated . The problem is in IG / HG compatibility in my opinion , because the chain is a little bit wide , and it has a little play along the teeth of the cassette. Do you have any information about the compatibility of this chain with Shimano HG cassette ?
That model number you provided does not end in the numeral "1". Rather, it is the letter "I" which stands for "interactive glide." That means that although it is of the Shimano HG model series it is only compatible with IG chain. IG chain has an external width of 7.1mm compared to HG's 6.6mm.
Your KMC Z8S chain has a outside width of 7.1mm. It is therefore in the family of Shimano IG compatible chains along with KMC's Z51 and Z72. What you may try is adjusting your B tension screw on your rear derailleur. Adjust it so that your upper carriage/gear pulley is close to the low gear sprocket but not hitting it. Doing so will improve chain to tooth engagement.
Also, check that your derailleur/gear hanger is not bent and that your derailleur limit screws are properly set.
If this doesn't work then your sprockets may be worn despite appearances. I have encountered freewheels/cassettes that appeared un-worn but exhibited the slipping you mention even after installing a new chain. The problem was solved upon installation of a new cassette. Slipping sensation may also be caused by a worn or damaged hub body (aka, freehub) as the pawls fail to properly engage the freehub teeth. In such case it isn't the chain but the freehub itslef that is slipping but it can feel like a slipping chain.
Lastly, if your chainline is way off you may have chain rub or slipping front or rear. If you recently changed the bottom bracket, serviced your rear hub, or swapped sprockets within your cassette cluster check that all spacers were replaced properly. Also verify your new bottom bracket spline width and offsets match the old.
IG parts are relatively hard to find to begin with, so I'd bet that something is not adjusted properly (or its in your head and you should just deal with it - there is a bit of play when you've gone through a chain or two on a cassette when you put on a new chain at first).
To quote Sheldon Brown:
"Some Shimano 7- and 8-speed cassettes carried the Hyperglide principle even farther, by adding ramps and contouring to both sides of the sprockets. To do this while maintaining the full thickness of the teeth, the designers had to make the sprockets thicker. Since the sprockets are thicker, the spacers must be thinner to maintain the correct spacing. Also, since the sprockets are thicker, Shimano says you shouldn't use H.G. chains on I.G. cassettes. SRAM chains are compatible with both types. Shimano says that you shouldn't mix I.G. and non-I.G. sprockets, but if you use one I.G. spacer per I.G. sprocket, you should be all right. Some experimentation may be required for best results with a mixed setup."
To quote this thread: "All current HG cassette cogs are specially profiled to be HG or IG chain compatible. Any Shimano IG chain will work with any HG cassette. If you use a HG chain with a 7-speed IG cassette (CS-IG50 and CS-IG60) the rear derailleur adjustment becomes more critical. One significant compatibility issue is always to avoid using a HG chain with an IG-only crank set. Why? The chain will release from the large (outer) chain rings too easily while in the larger rear cog. It may be possible that the chain drops off the outer ring without shifting the lever, when under full power."