Joining a pure strategy consultancy (those that do just strategy, as contrasted with the ex Big 5 that do a bit a strategy consulting and a lot of other types of consulting) is top of the list of most candidates' aspirations. Strategy firms charge the highest fees because they do the most interesting Board-level work, a benefit of which is that they can also afford to pay the most.
Consequently, there are often hundreds of applications from top candidates for each graduate entry-level position on offer. It is really only worth your while applying if you have a 2.1 or above from one of the top few universities in the country - which in the UK I would classify as Cambridge, Oxford, LSE, Bristol, Durham & Exeter. Some of these firms will only really look outside Oxbridge if they are concerned about not meeting their recruitment targets.
If you fall into this category, great! The candidate report you will find on Top-Consultant's homepage gives you a breakdown of some of the leading strategy firms. If you don't, I would suggest trying to get into the strategy arms of one of the ex-Big 5 firms. They are not quite as prestigious - and therefore the entry criteria are slightly easier to meet - but when you come to make your first career move in a few years' time, you may well be able to move to one of the best strategy firms if you have some form of strategy consulting on your CV.
Hope that helps and best of luck, Tony