I understand that cleaners are hard to find but my advice as someone heavily involved within the domestic cleaning industry would be as follows:
If you go with a private individual service provider, please make sure for your own security that they are registered to operate a legitimate tax paying business and are not committing benefit fraud or working 'under the radar' of HMRC because if you are trusting someone to come into your home they need to be honest.
They should have received documents from HMRC when they first set-up. Also make sure they have public liability insurance to protect you if they cause massive damage (fire or flood or if they break a large expensive item like a television then they are covered, these things happen!).
These two criteria are both requirements of any start-ups wishing to join our trade association as they are the hallmarks of a professionally and legally run service. You definitely don't want the 'other' kind and neither do we. One way to find out if a cleaner is registered as a legit, self-employed business is to NOT pay them in cash, anyone who wants cash only has a reason to ask this, they should be offering other methods to pay them too.
Cleaners are very hard to find by all accounts, hence my own cleaning business doing so well, and we are not cheap, but do not get so desperate as to hire someone dodgy, they will be coming into your home and touching and seeing all your possessions, the relationship relies upon trust both ways so take these steps to make sure you are protected and that you get someone who's intentions are above board.
I don' t think it's up to you to do background checks as you are not his/her employer he/she will be self-employed but they should be meeting the above criteria.
After that it relies upon whether he/she has room on her schedule for a new client as most good individual cleaners become fully booked within a few months.
Also ask about their service level ie what is provided within the price quoted and what is additional to pay for, see if their service matches your needs.
Good luck, I hope this one is a good match for you.