to answer your first question, whatever is your current experience (Vlad provided a good list of the possible options) the best way to break into consulting for an experience hire is to network and apply via referral. You don't need to attend events necessarily to do networking; in particular, if you are unable to attend you can follow the steps below:
1. Identify the ideal connections to contact. LinkedIn could be a good starting point for that (another option is to use your university alumni database, if available). Former alumni of your university normally represent the strongest connections.
2. Write a customized cold email. I would recommend using emails rather than LinkedIn messages, as writing via LinkedIn you would immediately be perceived as a stranger (there are ways to easily find the email of everyone nowadays in few minutes). The email should follow some specific rules for the subject and the content, in order to maximize conversion.
3. Have a call with the consultant, and indirectly ask for a referral. The call should be structured in order to maximize conversion, as the email. You should have an idea in advance of (i) how to introduce yourself (ii) which questions to ask and (iii) how to ask for the referral indirectly. In case you manage to get an in-person meeting instead of a call, that’s great but not strictly necessary.
I have used this method when I applied for consulting, and I believe it’s the main reason why I got six invitations out of six applications by consulting firms (MBB, Oliver Wyman, Roland Berger and former Booz&Co - now Strategy&). I never applied through the website only, as I always had a person referring me.
As for your second question, the likelihood of being hired as experienced hire will depend on
- your background (how much do you fit with the company core practice) and
- the size of the company in your country (in general the bigger, the easier, as there are more slots available).
MBB are the biggest strategy consulting companies in many countries, however, since they are also more competitive, they may not be necessarily the easiest to land an offer for an experienced hire.
Hope this helps,