How to convert internship into full-time-role?

Hi, 

I will be doing an off-cycle internship starting in February. According to the interviews I was in, the firm offers approx. 20% of interns a full-time offer. Given that they only offer full-time-roles for only such a small percentage I wanted to ask you what, in your opinion, is the most crucial thing you are looking for.

Aside from delivering my work on time with as few mistakes a possible, is there any other way to positively stand out? Also, do you look for interns who are eagerly motivated and that come up to asking for work or do you prefer that you approach the intern. Also is it good if I ask you various things like "what other projects are you working on", "how can I improve" etc, or do you prefer being leaved alone since your work is stressful enough as is?

Thank you!