Hello! Single lady trying to conceive here. I am primarily having treatment (IVF - I'm a few years older than you) through a clinic, but have had a few tries at home insemination via Cryos as well. (Actually, there's a big yellow box of sperm sitting in my spare room right now, waiting for me to ovulate...)
So. My advice would be, firstly, go get some basic checks and tests done via a clinic. Primarily, these would be to look at whether you ovulate, whether your tubes are clear, and what your egg reserve is likely to be. This will set you back a few hundred pounds, but it's really important to know these things - home insemination can theoretically work and is (a bit) cheaper than clinic treatment, but it'd be a massive waste of money if you spent thousands of pounds on sperm and it turned out that there was some impediment to you conceiving in that way. I wouldn't necessarily tell the clinic you're considering home insemination - they may be a bit disapproving! - but maybe just that you're considering your treatment options and want to find out where you stand (which is true).
I would actually encourage you to seriously consider clinic treatment rather than home insemination anyway - you'd have access to a wider range of treatments, and would have better odds per round of treatment. If you look on the HFEA website you can find success rates for women in different age groups, for different treatments. I know I was really surprised by how low the success rates are for IUIs (the method of insemination used in clinics) - I just sort of assumed it was relatively easy to just, as it were, whack some sperm up there and Bob's your uncle, but apparently not! And the rates for home insemination should be assumed to be significantly lower still. I ended up opting for IVF via my clinic, largely because the success rates are so much higher. (My forays into home insemination are largely wishful thinking - I keep hoping it might just improbably work! - and also are psychologically helpful in maintaining a sense of things moving forward in between my IVF rounds).
If everything checks out and you do want to try home insemination, timing timing timing timing timing timing timing. I'd recommend spending AT LEAST three months tracking everything you can about your cycle - cervical mucus, temping, OPKs - and building up a picture of when you ovulate. Again, this is something where in my opinion not doing it risks wasting lots of money through poor timing. Frozen sperm doesn't live anything like as long as fresh, so your window of opportunity is much smaller than it would be doing things in the old-fashioned way. You really need optimal timing, which takes a few months to begin to understand.
Finally, don't underestimate how expensive Cryos can be. You tend to need to use more than one straw per insemination, and ideally you probably want to do more than one insemination around your ovulation, to maximise your odds. Also, it's not immediately obvious but there's 25% VAT on top of the prices. So, it can all add up.
I hope that was some help - feel free to ask if you have questions!