I (French citizen) started a PhD in computer science in the US one year and a half ago: what can I do to get the US citizenship as quickly as possible?
There are several options:
Marry a US citizen. Once the US government is satisfied that your marriage is not a sham marriage (i.e.: you didn't get married just for the immigration benefits) - you'll get a green card (permanent residency). After three years as permanent resident married to a US citizen, you can apply for US citizenship.
Get a green card on your own. Either through an employer (EB1/EB2 process) or because you're such a distinguish scholar that the US would want to keep you (EB1 self sponsored process). After 5 years as a permanent resident you can apply for US citizenship.
Have a congressman introduce legislation to the US Congress to grant you a citizenship. Once the bill passes and is signed into law by the President - you'll become a US citizen.
Join the US military. After 4 (IIRC...) years in the military, as long as you're not dishonorably discharged, you can apply for US citizenship.