First, the obvious: You have to do good work. But how do you know what good work looks like to your new boss on a new job? Easy. Ask. When asked during your interview if you have any questions, one of your questions should be, "What are your expectations for the first few months on the job? What do you see as the highest priorities, and what would a successful new employee look like to you?" Note down the boss's answer, and do exactly that when you get hired.
Second: Look at your boss as a sponsor and mentor, and also look around the organization to find someone at a higher level who can sponsor you and guide your advancement within the organization. The fact is, few people advance within any organization without making friends with people above them. Even better, a particular sponsor or mentor already knows the ropes of the organization at a higher level and can guide you in becoming successful at that higher level.
Note that I am not talking about brown-nosing, although some people would call it that. I'm talking about simply making friends and being a friend to those above you. If you are bright and capable, you will be noticed, and your career will advance. It will advance more rapidly if you find a sponsor, someone above you who is already successful, to be your guide. Look for somebody with whom you have a natural affinity, someone with a personality and interests similar to yours with whom you will feel comfortable developing a friendship.
Also be aware of company politics as you pursue friendships at higher levels. Your boss may feel threatened if you make friends above him or her, so it's important to include your boss in your circle of friends and mentors--do NOT bypass him or her, as that will certainly lead to trouble. In general, you need to make friends with everybody you work with and avoid alienating anyone, especially difficult people. You will always have your critics, since that's just a part of life, but you can minimize criticism and political difficulties by being kind and friendly to everyone.