You can show without being a club member (though there are some of the duplicate classes that are only open to club members e.g. novice, intermediate and FMC/MPC). You can only become a member of a club, e.g. Southern, Midland or Northern, not a member of the National Hamster Council. The NHC is the overarching organisation that the clubs run under, so being part of one of the clubs, you are part of the NHC already
There is only one show run by the National Hamster Council and that is the Bradford Championships aka the Burgess Premier Small Animal Show held in Harrogate in January. All the other shows are run by one of the clubs and you can show at any.
If you are a full member of one club, you can then be an affiliate member of another club. For example, I am a full member of the Southern Hamster Club (i.e. the SHC secretary sends me my journal and I have voting rights at the AGM, can be elected to an official/committee post for the SHC, and enter the Members Points Class). I am also an affiliate member of the Midland Hamster Club and the Northern Hamster Club. This costs less than a full membership and entitles me to vote at the MHC AGM, likewise be elected to a post with the MHC and enter the club specific class called 'Free Membership Class'.
I'm not sure about the Northern rules as I've never made it to the Northern AGM!
I'd suggest starting with a club membership of one of the clubs and, if you find yourself attending a lot of shows run by another club, consider an affiliate membership of that club also