London to Liverpool for two on a budget

Firstly you can get discounted Train tickets ahead of time if you're willing to commit to particular trains. These are usually on sale about 12 weeks ahead and the cheaper ones will sell first. Train takes about 2 hours 20 minutes each way.

There are two Beatles Museums in Liverpool, both costing £20 with concessionary prices. The Cavern Club is free to enter.

To see other Beatles associated sights (Penny Lane Strawberry Fields etc) you will need to get to them. You can get a taxi tour but that is typically around £160 for up to 6 people, so very expensive it oil's one or two of you. You could I guess organise your own cab to show you some specific sights. The National Trust to McCartney's and Lennon's childhood home costs £37 and must be booked well in advance.

I've just been looking at the Cavern Club website and it says it's £5 to enter, so I don't know whether I'm misreading that.
Anyway I've just been looking at The Magical Mystery Tour which costs £21.95 for a 2 hour tour.
Is that expensive?- a subjective question.
Doubtless you could do it all by public bus- buy a plus bus ticket with your Train ticket for £4.20 a day or £2.75 with any national railcard.
It will take some planning to work out an efficient route and take a lot longer than 2 hours.
There are other companies as well but in cost terms I suspect they are all much of a muchness.
Whether or not £22 is a lot there is an inherent convenience factor to take into account.

I bet the tours are all basically the same, look at several of them and see if you're comfortable copying their itinerary on their own.

If you're talking about a daytrip, you need to work out the timing on that too. Looks like the train is about 2.5 hours each way.