Breaking into VC/PE from undergrad is very challenging. I did it coming out of university (I cannot reveal which university since that will give me away...I'm on my firm's website). I also cannot reveal which VC fund for similar reasons, but I will tell you that it is an independent (not bank arm) and top-tier VC based in Silicon Valley.
In my case, I offered to work for free over a summer. In the end, they ended up paying me which was nice. You'll see that most professionals are not cold-hearted and will be nice if you show the willingness to work hard. I would recommend this if you do not have financial difficulties as it shows real dedication (no risk for the VC fund, serious candidate). I had some entrepreneurial experience in digital media, and spent the previous summer working as a business analyst in the technology group for a commercial bank (i.e. evaluating end-user requirements for custom-designed software).
I know a lot of people say that you should never work for free, but in my opinion, as an undergrad student your degree is commonplace so I don't see any problem doing so in order to break into a competitive field. Even if you're from an ivy (like me), the competition to break into PE/VC is fierce. After the summer, the partner who I worked for really liked me and pushed the other partners to create a new analyst position at the fund.
Feel free to ask me any further questions. I've talked to various VPs/Associates at my fund regarding recruiting new hires as well. So if you're interested in breaking into the industry I may be able to help you out.