As someone who has worked in the staffing industry, it is ok to let the client know that you enjoy working for them. HOWEVER...proceed with caution. First, why are you in this seat? Are you covering a medical leave? Are they short handed because of a major project that will be ending soon? Is the position vacant because of turnover? You don't want to ask for them to consider you unless you know that they are prepared to hire. If they have a hiring freeze, don't press the issue because you are wasting your breath.
I recommend to let them know that you are happy with the position, and you would like to find ways to help them even more. You don't have to tell your agency to give your 'boss' the message. Make sure you are on point all day, every day. The last thing that you want to do is to look like a 'temp' Don't take personal phone calls, surf the web, or get caught gossiping in the lunch room. Employers are watching!!!!
I also recommend to look at any postings the employer has that might fit your skills. You are always welcome to apply, but do keep your agency informed, as they might have an agreement with fees, length of assignment, etc. Being professional and patient is always the best way to go in this economy. I don't usually see temporary candidates get hired by the client until they have been on assignment for at least 6 months.
excellent post bravo