Now, if by "hacker" you just mean "developer" or "software engineer," then one could be a little bit more detailed, but there's still a lot of different areas to explore.
That said, if you want suggestions for books, articles, links, etc:
I'd suggest you start out by reading some books / articles that give a bit of the history of "hackerdom" and some insight into the hacker mindset. Some possibilities that come to mind:
Hackers & Painters by our very own Paul "pg" Graham
Hackers: Heroes of the Computer Revolution by Steven Levy
The Soul of a New Machine by Tracy Kidder
Cyberpunk: Outlaws and Hackers on the Computer Frontier by Katie Hafner & John Markoff
Artificial Life: A Report from the Frontier Where Computers Meet Biology by Steven Levy
Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter
Most or all of Paul Graham's essays. http://paulgraham.com/articles.html
Read some of Edgar W Dijkstra's works: http://www.cs.utexas.edu/~EWD/
Couple that with a good book on the C language, some basic "theory of computing" stuff and a good algorithms book. I actually like Java as a good language to explore a lot of different things, since it has great library and tool support, plenty of community support, plenty of docs and books, etc. Learning some Java so you can play with cool libraries like Mahout, OpenNLP, etc. isn't the worst idea in the world. And, of course, learning a Lisp of some sort. This is still on my personal TODO list as well, and I've been dabbling in Clojure... but I don't think you'd go wrong with Common Lisp or Scheme either.
Other book recommendations:
Code: The Hidden Language of Computer Hardware and Software by Charles Petzold
The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine by Charles Petzold
Structure and Interpretation of Computer Programs by Harold Abelson and Gerald Sussman
Hacker's Delight by Henry S. Warren
Operating System Concepts by Abraham Silberschatz
Introduction to Algorithms by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein
One (or more) of the Algorithms in FOO series by Robert Sedgewick. There are Algorithms in C, Algorithms in C++ and Algorithms in Java to pick from. Each is a multi-part series. The Java one goes up to at least 5 parts, I can't recall offhand if the series numbering is identical across languages.
I also find that most hackers have an affinity for F/OSS software. If you don't already, I'd suggest running Linux or BSD as your operating system. Learn the standard command line tools, learn at least one of (Emacs|Vim).
Read The Cathedral & The Bazaar if you haven't yet.
Also consider some of esr's various rants, some of Stallman's stuff, etc., for context if nothing else.
Edit: Also, expect somebody in this discussion to recommend reading The Art of Computer Programming by Don Knuth. I'll just say that TAoCP is more reference material than tutorial, and isn't something you'd normally just "read" as in "read cover to cover." It's a great reference to have around for when you need to dig into the deep details of implementing an algorithms, and need the mathematical stuff, etc. But it's not something you necessarily want to start with.
There's SO many more places you can go though, than all this. The stuff mentioned above is good foundation stuff for a lot of paths you might want to explore, but don't limit yourself. Buy and Arduino and start doing some hardware hacking, read chemistry books and biology books, whatever floats your boat. Like ChuckMcM says below, curiosity is huge. If you're curious and don't limit yourself, you'll be a hacker.