I will become a “contractor” to the US based company and they will deposit my check into a UK bank account. I understand that my paycheck should not have US taxes (fed, state, ss) taken out. What might my UK tax rate be and do I have to register as self-employed?
Promoting comments to an answer, but with a big caveat - you need to talk to a proper advisor about this!
If you are only working for one single company long term, you are unlikely to be able to do that as self-employed in the UK. The term to google for is IR35. Basically though, with one long-term company / client, you'll be deemed to be effectively an employee not self employed, and therefore owe the additional employers taxes.
If your US company has any kind of UK presence, you'd probably need to be employed through that.
If not, you'd likely have to pay the employers taxes directly yourself, which is quite unusual in the UK. The term to google for is DPNI (Direct Payment of National Insurance), start somewhere like http://www.hmrc.gov.uk/gds/nic/work/embassy.htm for guideance.
There are 3 main kinds of UK employment taxes - Income Tax (normally done as PAYE = Pay As You Earn), Employers National Insurance (paid by the company) and Employees National Insurance (paid by you, normally taken off by your company and sent direct to the government). You can use official tools like https://www.gov.uk/guidance/hmrc-tools-and-calculators and https://www.gov.uk/check-income-tax to estimate what those will be, to get an idea of your effective tax rate
(Since you'll likely have a complicated situation, including things like earnings and interest from overseas, you'll likely also have to do a UK Self Assessment tax return once a year.)
You'll need to ensure that you're paying / not paying the right taxes on the US side, along with ensuring that you're correctly signed up for paying the UK taxes at the right time, including UK taxes that people don't normally pay themselves because their companies normally do it for them. That's why you need a proper advisor to ensure you do it all right!
Oh, and don't forget that you'll still have to do US tax returns as you're a US citizen, and you'll likely also have to tell bits of the US government about overseas bank accounts, overseas shareholdings if you have a UK pension etc. We have some questions on those things here, check those too