As you might have guessed, the answer is: it depends... size of the firm, location of the firm, area of law, law school attended, grades in law school, extra-circulars (law review, legal clinics, etc.), and other factors like judicial clerkship.
"Private law firms" can mean anything from the local village attorney who practices lots of types of law, helping people with wills and house closings and disputes in local court, to the AmLaw 100 Firms, pulling in billions of dollars a year in revenue and representing Fortune 500 corporations (and ultra wealthy people). Moreover some markets (NY, SanFran) may be more competitive than others. And some areas of law may be "hot" - which changes, as this ebbs and flows with the economy (e.g., recessions mean more need for bankruptcy lawyers). Of course, the higher "tier" law school an applicant attends, and the better ranked (higher GPA) they were as student, then the more doors open.
As you can imagine, the big firms want applicants from big, top tier law schools, and they want students at the top of their class. Those are the toughest opportunities to secure. They often pay the most in starting salary (and have the most grueling hours, too). Smaller and mid-size firms want the best talent they can get too, but they may be more willing to consider factors more than just law school tier and rank. At these firms you might get more client facing - and litigation - experience sooner than if you are working under the strict partnership hierarchy in a big firm.
One big advantage is to try to get a summer associate positions (akin to a paid summer internship) at the firm you are interested. Summer associates get to audition the firm they might work at - whilst the firm auditions the student. Successful summer associates will get an offer to work at the firm the next year (after they graduate and sit for the Bar Exam) - so it is a huge relief going into the last year of law school to know you have a job lined up. However, you need to get great grades and hustle to line up those summer positions!! (Summer associates opportunities are more common at larger firms.)
My best advise is go to the best law school you can and get great grades while you are there, start your job search early (visit the career office your first semester to make a job search plan!) and hustle!
Desiree recommends the following next steps: