There are many ways to get a full time job , and it has its own benefit and disadvantage.
I personally like to work as an contractor as it means higher pay, more diverse experience and more opportunity to learn new skill.As I am single and not at a point to settle with a family , its works better to work as an contractor.Most important for me is peace of mind and zero politics.
If you want to get inside a company as a direct employee , look for contract to hire clause in your contract. This is the first step .Make sure you don't have any clauses that wont let you transfer.

Second, become essential part of the team. Somebody who can't be replaced easily in short time. One example to create a dependency is to develop your own framework or software and implement it as essential working tool.

Develop some leadership skills, you will never be hired by your client if you don't show any leadership.
Wait for right time to apply, look for openings posted on intranet. Never reveal your intentions.
Always remember if the position is higher there are inhouse people applying for it. Make sure you don't compete with your boss or you may even loose your contract job(if you are in corporate culture )
You will go through interview process but as you have gained knowledge about the culture it would be your plus point against outsiders.
Thats all..its not only one way there are many ways people get directly hired.Some don't even have to go through interviews.
What's your situation???