How independent should an academic be?
Collaboration is generally seen as a major strength in computer science and computational science, not a weakness.  One of the biggest questions we ask about any faculty candidate in my department is "Who in the department/college will they work with?"  We do occasionally hire faculty that work alone in their cave and occasionally emit theorems, but they're rare exceptions (and even they still work with their students).

The most important thing is to establish a reputation as a leader in your collaborations, rather than a follower.  You need to establish your own research agenda, instead of just following someone else's.  More importantly, you need to be seen to have your own research agenda, instead of just following someone else's.  Your agenda must be visible in your publication record, in your recommendation letters, and (eventually) in your funding record.  To this end, it is very important to do a few things:


Limit your collaborations with more senior researchers, especially your advisor, especially between getting your PhD and getting tenure.  As I've suggested elsewhere, publishing at least one paper without your advisor before you graduate is a strong signal of independence, even if you have other coauthors.  Once you have your PhD, DO NOT publish with your advisor for at least a few years.
Do not always publish with the same set of coauthors.  It's fine to have two or three different groups of people that you always work with, as long as you can wind a consistent story that ties most of your work in those different groups together.
(Once you have a permanent job:) Get at least one grant as a principal investigator, not just as a co-PI.  Definitely join other grants as a co-PI, too, but you must have at least one grant with your name on top to get tenure.