They have all got their pet cleaning firms already. You can only hope to get on their books if you are cheaper, more reliable, available at short notice.
I find the smaller firms better. We got the cleaning contracts for a larger firm who basically owned us for a while and then dumped us without a phone call for someone cheaper.
To approach them, its best to put on your cleaners uniform with the comapny name on, if you have such a thing, ask to speak to the letting managers, tell them about yourself and leave your details (leaflet/letter/card etc).
You will get a good reception from mostly everyone, but don't translate that as being on the books. They are always on the lookout for cleaning companies, but tend to look at costs. Some might ask you for your menu of costs.
You also need to be able to clean to an inventory check-out standard.
Don't expect the phone to ring straight away. You are more likely to to get a call if a their cleaners are fully booked/let them down/or need it yesterday.
Follow up with a courtesy phone call to see what your chances are.
I've done this a few times. I spend a day every so often going round all the letting agents in a town. Sometimes you get one - sometimes you get none.
You need to go in to see them for a variety of reasons (especially to find out who their decision-maker is) but you need to talk to them to find out what they need, what they have got, and what bugs them.