Patch ice is the session which figure skaters use to practice on and have lessons (in the US they are called freestyle). If you are having your lesson on a patch session, you need to pay the £4 entry for the session as well as the cost of your lesson. In group lessons, the ice time is paid for in the cost of the lessons but this doesn't happen for private lessons which is why it is an extra cost.
At the moment, the only other cost is blade sharpening but you don't need your blades sharpened every week (I skate about 10-12 hours a week and have my blades sharpened every 4 weeks or so). You can join a club if your rink has one, which will probably have a joining fee, but that is optional.
Eventually, if you want to test the NISA standard levels or compete you will need a membership and there will be other fees, but at the moment your only weekly cost will be for your lesson and patch.
In some rinks you can't take a private lesson during a public session, but if this is allowed in your rink and you can find a session that is quiet then you might prefer to have your first few lessons during public so you don't have to worry about other figure skaters so much. If your public sessions are busy or you would rather skate on patch then that's fine too - don't worry about what level the other skaters are! They were all beginners at one point and should be looking out for you. At my rink, there is a girl who is about 9 or 10, and her first private lesson on patch was the first time she'd skated. She's been taking lessons for a couple of months now and it's been lovely seeing her progress and get more confident. All of the higher level skaters keep an eye out for her because she hasn't reached the level where she is always aware of where other people are or what they are doing, and the other girls her age seem to get on with her well. There will always be those not-so-nice people who will think you're not "worthy" of private lessons until you reach a certain level, but it's up to you when you think you need them. I started private lessons after passing level 6 because that's the point where I felt private lessons would benefit me more than groups. Other people will only take private lessons when they have finished the Skate UK course and can't take group lessons any more, other people start private lessons from level 1 - it can help a lot to have the individual attention but everyone is different, some people do better in group lessons, some in private lessons.
I hope that helps you :)