hi nofel,
i think in general, as long as you are not asking for money, that is, you are just applying for a place as PhD student but not for funding, it is not too hard to get in. they like having lots of paying students. just fill in the online application form. if you have good grades and a project which matches one staff's interests, you have good chances.
however the LSE is not very good at providing funding. so if you are applying for a funded place, you might find that competition is fierce. in my department there are 2 ESRC funded places/year and perhaps one or two people who get "full" funding from the school/department, compared to about 22 who start every year. and that's not counting those who don't start because they applied for but didn't get funding.