Xenia, as ever, is right. And those aiming for the big City/Accountancy firms will need to start early and devote a fair amount of time to it.
Start early. Many Universities will be hosting various events by employers. At LSE these were often in the form of breakfast talks by banks' economists etc. Wear a suit (or pink dress), bring along a CV and get a free croissant. Apparently at Oxbridgeit was dinner. Book a session with the careers office early. Then apply to as many as you have the time for, as lots and lots apply to each. Each will have a lengthy process, often involving application form, on-line test, Skype interview, further competency testing and an assessment centre.
I think it is worth taking a step back and considering where the weaknesses are. DS had no problem with the competency side, but really struggled with the Skype interviews. Not eveyone is a natural interviewee, and it is worth looking for tips and tricks, either from the careers office or the Internet. It is also worth making sure that any academic referees know who you are, which can be difficult on a big course.
Some employers will do up to 80% of recruitment from internships. And they can be a way of working out what you don't want to do. DS' peers tended to be very employment focussed, right from the outset. I don't think this is as true elsewhere. (In truth it was a bit scary.)