Build rapport with the people you are working with, if at all possible. Make them respect you as a professional and like you as a person. If they have the option, they are much more likely to keep you on if they like you as a person.
Learn the Ropes
One of the pains of having to hire new people is having to spend a whole bunch of time training them on the ins and outs of the company. Procedures, shipping policies, copy machines--even the smallest things need to be shown to brand new employees. Learn the ropes quickly and make yourself invaluable. They may be able to see the advantage of keeping you on rather than having to worry about hiring someone else.
Most employers have pretty low expectations when it comes to temporary workers. Don't adopt the role of the temporary employee--do as good of a job as you would if you were a regular worker and were coming up on your performance review. If you shock and amaze them with your amazing career abilities, they may be more likely to keep you on.
Explore the Possibilities
If you like your temporary job, talk to the HR department or your immediate supervisor about the possibility of being hired on into the company. Look for open positions and see where you might be able to fit in. Even if you can't get the exact same position you are temping for, look for other ways to get your foot in the door at the company.