Self-employment and business tax deductions

Here's my situation:

In February I left a job that had been paying me ~£22,000, paying tax and NI through PAYE. I was paid my March salary. Since then, I haven't been working and haven't claimed any jobseekers or any other benefits. At the beginning of October, I started working as a contractor for two companies based abroad, one in Spain and one in Italy. It is remote work, internet-based. I'm starting slowly but fairly soon I should be earning only slightly less than I was at my old job. I've been advised that I need to register as a Sole Trader and I have begun the process for that.

I believe that since I wasn't working for the first six months of the tax year, my earnings will be below the minimum threshold and I won't owe any tax for 2016. However, I have a few business expenses, namely a new computer, that I think I should be able to deduct. My questions are: Is the tax-free threshold the same for self-employed people as is is for regular employees? And if so, is it possible to claim a deduction for business expenses in a year when I don't actually pay any tax?

I know very, very little about the UK tax code so anyone with any knowledge feel free to talk to me like I'm a slow child, I really just want to know what I have to do to be compliant whilst not overpaying, and the uk.gov website is typically opaque. Thank you!

The Personal Allowance is £11,000. The Personal Allowance is deducted from all types of income, including both employment and self-employment income.

The information required in the self-assessment return is limited. The form that applies is SA103S. The details will be- sales in box 9, total expenses in box 20, profit in box 21, the cost of the computer in box 23 (described as annual investment allowance) and the taxable profit in box 28.

HMRC will calculate both the income tax and National Insurance liabilities. 

You will need to file your US taxes as a "sole proprietor" as well and the limit for doing that is a net income of $400. Make sure you avoid paying the US self employment tax.