Here is a short video demonstrating how to clean carpet grease stains. The presenter is a carpet cleaning instructor for professional cleaners. This is an excerpt from his video series. Grease spots are typically found around entry ways, near the front and back doors, near the kitchen or by the garages. He uses dry solvents to moisten the stains and then them up with a cloth.
He then completes the cleaning process by extracting the fluids. In the video, he uses a large professional grade carpet extractor (carpet cleaning machine). To emulate the process at home, you may want to look at these types of stain solvents.
Using a home carpet extractor would be helpful, but you could also try to blot the fluids out.