This question has been asked a lot in the past but so far there is no statistical number to prove how many graduates or employees from non-business background landed a job in the consulting industry. Indeed, management consulting is a very attractive career - even John Legend spent a few years working at Boston Consulting Group (BCG) after graduating from UPenn. According to insiders, there are a great number of consultants whose majors aren’t business or engineering related, which (lightly) shows that a business background is not really the top priority of consulting firms. However, the firms require you to understand some basic business terms (assuming that you apply for entry level), which can be found or purchased on many management consulting prep websites, eg: mconsultingprep, etc. The most important factor here is your business mindset and the firms will test you through case interviews to evaluate how you solve a business problem intuitively. On the other hand, management consulting involves a lot data reading & analyzing so you also need to practice hard on math skill and learn how to explain things clearly.
Upon applying for the job, it is easy if the firms organize on-campus recruiting sessions at your university and you can apply in the standard way for an internship or an entry-level position. Nonetheless, if you're a non-business major at a non-targeted campus, you have to be more flexible to get into the case interview round, because this round is when you can meet the senior consultants and every interviewees have to kill it the same way. Make use of any possible connections from your family, friend, colleagues or alumni who have worked in consulting or know people who work in consulting. Pound the pavement & exploit your networking circle as hard as possible to meet up with some in the field and show them your interest. If they like you enough, they might send your CV to HR and there’s your first baby step. Prepare your CV well because in hundreds of thousands of CV, everyone is as good as you so good contact at the firm you apply is an advantage.
After that, it all comes to practice, practice and practice if you apply for prestigious firms like Mckinsey or BCG because they have the McKinsey PST & the BCG Potential Test to evaluate your “data interpretation" & “critical numerical reasoning" skill (aka math test). Then if you pass this paper-based or computer-based test, you’ll get into the in-person interview, which is popularly known as the case interview. Again, you don’t need to have experience to overcome these but it's about finding the right materials and building the right study plan for your recruiting process. These days, there are many sites offering both free & paid materials for anyone who's interested in management consulting so the options are quite diverse. As far as I know, MConsultingPrep is a big & reliable name for beginners because they offer both math drill tests and case interview question packages for both Mckinsey PST & BCG Potential Test so feel free to check it out (plus: to be honest, I find the price more reasonable).
Anyway, good luck folks!