what is the best way to do a freelancing job over the summer for a student?

I've had a similar experience in Canada.

Here I was able to work self-employed at any profession and earn up to $15,000 in a year before having to charge customers taxes.

As others have mentioned just keep good records of your incoming money and claim it as income on your taxes.

As a student I don't think you will need to pay tax however you will need to register as self employed. This is a simple phonecall to the inland revenue who will send you the information you need. You will have an extra national insurance contribution but it is small. Even as a non student your first 5k or so, is non taxable. If you did have to pay tax, you would want to make sure you have copies of receipts for work expenses as you are taxed on 'profit' not earning (ie after expenses)

freelancers do not need a company to work as. You can treat it as an extra job , and can be taxed accordingly based on your local laws,your income might not be sufficient enough to be taxed so unless you are continuing as freelancer and earning much more than this per year, you might not need to do anything except do the work and get the payment. You will need to know how to receive money , at least should have a bank account to get the money.

Easiest thing to do between the US & UK is to do bank wire transfers for money.

I'm not familiar with UK tax laws but in the US all you would need to do is make sure to claim it on your taxes at the end of the year.

That's the minimum and easiest.

People create a company if they plan to do work for many clients, or want to protect themselves more from potential liability. If you are working on stuff where you may get sued then it helps to try and separate you personally from your business entity.