To join, you need to be 18 and have a UK credit or debit card billing address. Fill in the form on the TfL website, select what kind of access you want (24 hour is £1 + usage, 7 day is £5, or annual is £45; full cost breakdown here), and complete a few other personal details. You'll then be sent a key in the post (each key costs £3) to access the bikes at the docking station; see the map for the full list of stations about town. The bikes will be available to use from July 30th.
This registration is for members' access; casual users, i.e. those who enjoy simply rocking up to a station whenever they like and hiring a cycle on the spot, won't be able to use the bikes yet — a vague note on the website says that they'll be available 'soon'. Perhaps Serco (the group operating the system) want to get any teething troubles out of the way by opening it to early adopters who, by their very nature, are more likely to be familiar with cycling in London.
Fancy owning rather than renting? Try this new way of buying a second-hand bike