If a US citizen marries a visiting UK citizen in the US would the UK citizen become a US citizen and what would he have to do to become one?

No, the UK citizen would not become a US citizen. First the UK citizen would have to apply through INS for a work permit and perminent residency. The work permit will be issued within month (probably), but perminent residency could take 18 months or more; during which time the applicant would be expected to remain and work in the US (short overseas visits sometimes permitted). Once perminent residency has been established a 'green card' will be issued and 3-5 years after this the immigrant can apply for US citizenship.