I work US based company from UK. Do I have to be self

Hello and thank you for allowing me to assist you.
If a U.S. company with no U.K. presence (entity, representative office or branch), hires a U.K. local as their first U.K. based employee, working from home, there is no "taxable presence" (i.e. no business address of the company) in the U.K. and therefore the U.S. company will typically not need to operate PAYE withholding.
Instead, the individual will be required to operate a "self administered" PAYE arrangement, known as a "Direct Collection" payroll to settle U.K. income tax and Employee National Insurance (NI) Contributions (although Employer National Insurance Contributions are not due). In practice, most U.S. employers will not want to burden an employee with this administration and will instead use a U.K. based agent to operate a standard PAYE withholding scheme, even if this is not strictly necessary.
Under the same facts as above, but there is a "U.K. office address" of the U.S. company, U.K. PAYE is required and must be administered by the U.S. parent company.
If you are a US person and the company has sent you to the UK a "taxable presence" exists and PAYE withholding will be due and must be administered by the U.S. company - even if the you continue to be paid through the U.S. payroll. NI Contributions can be avoided by obtaining a Certificate of Coverage from the U.S. Social Security Administration, provided you do not intend to remain in the U.K. for more than 5 years.
It really depends on if you were sent to the UK or you were already in the UK.

Self assessment.If your studies are building on an existing skill, rather than providing you with the ability to attract new clients with a new skill, then its an allowable expense.

The last tax year started on 6 April 2014 and ended on 5 April 2015. This would include the earnings from Oct 2014 to April 5th, 2015. You should register for Self Assessment by 5 October 2015. The date to file depends on if you file a paper return or online. Midnight 31 October 2015 if you file by paper and Midnight 31 January 2016 if you file online.

Yes you will be assessed the tax when you file.You are most welcome.My goal is to give you excellent service.Your positive rating is thanks enough

You stated you are an employee. The US company is really supposed to set up withholding. If you are not an employee then you can just claim self employed.

I suspected they had not, that is why I advised about the differences when a US company has an employee in the UK.You stated you were an employee. If you are self employed and not an actual employee then you can file as self employed.