It seems to me that you have two possibly conflicting goals: one being a short-term goal while the other is a long term objective and each has its own route to success. Based on your question, you want to:
- Increase your current pay (short-term goal)
- Become hired directly by the placement company (long-term goal)
So let's take them one at a time:
1. What do you do as a contractor when your role changes?
If you are a contractor then you are not paid directly by the placement company but rather through a second party contracting company. That contracting company (Party 1) creates a contract with the placement company (Party 2) to fulfill certain work for a specific amount of money (Contract A). They then go out to the market and find you (Party 3) and create a second contract to fulfill said work (Contract B). Note that the pay rate in Contract A will never equal Contract B... the contracting company makes it's money by ensuring the rate of Contract B is smaller.
Depending on the relationship between the contracting company and the placement company, a change in role could constitute as a violation of one or both contracts above and require an amendment to those agreements. Most certainly if you were hired with the expectation that you would perform work at a specific rate and are now being asked to perform work that is worth more, one of the three parties above is making out like a bandit while one or both others are left in the lurch.
How to handle this dilemma could become slightly political, however, as your interests will often not be in line with either the contracting company or placement company.
If increasing your immediate pay is you number one focus, your best bet will be to discuss with your contracting company the change in your duties. They will then likely renegotiate Contract A which will then give them leeway to increase the money you are making in Contract B. Depending on the placement company, however, this could possibly rub their HR department the wrong way (after all you are now costing them more money). I probably wouldn't let that prevent me from having this conversation though as for all you know that renegotiation of Contract A has already taken place and you are the one currently in the losing position.
2. How do you move towards being hired by your placement company?
If you want to move towards being hired by the placement company you need to very much impress them with your work (which it sounds like you are doing). You also have to develop good relationships with the placement company managers, express your desire to become a direct hire and be somewhat fortunate (read: dumb luck) that opportunities in department finances will turn in your favor.
In the meantime, however, grab up as much work as you can stomach. The best way to show that you are invaluable is to make yourself invaluable. That assumption of extra work builds up your reputation and ultimately that reputation is what will get you hired on full time.
Depending on your relationship with your placement company manager, you might also consider having a conversation about your future goals and desire to increase your responsibilities and pay. There is nothing wrong with expressing the desire to move upward and to pursue greater opportunities. If worded correctly you can show yourself to be ambitious (which is a very desirable thing in a company). This conversation can also have the affect of lighting a fire under a hiring manager as if you have truly made yourself valuable they may extend an offer simply to avoid losing you.
One last thing to keep in mind is that most contracting companies include in their contracts (Contract A) with the placement company something called a "buy out clause". This defines an additional cost for a placement company to buy out a contractors contract (Contract B) and directly hire them. You may want to review your current contracting company's policies to determine how large this cost is as it could play a role in any negotiations you hold.
At the end of the day how best to play this one out will depend on both the contracting company and the placement company. I would probably recommend a professional approach: work as hard as you can, build up your reputation and talk to the managers of both companies and be up-front with your career and salary goals.