On websites like the LLBean one mentioned above, that routinely deal with international orders, just fill in your address and debit/credit card details exactly as normal - they will ask what country you are ordering from. Many sites also take PayPal. You'll usually be offered a choice of different classes of international postage: very fast, standard, etc
When it comes to UK customs duties, with some sites you'll get an email or a card in the post, usually quoting your order number or details of your order, telling you how much to pay the Post Office, who will be collecting the duty on behalf of the UK government. You pay, usually over the phone or online. Then your package will be delivered.
Sometimes the email or card will be from one of the 'big name' international courier companies, who are collecting customs duties and doing the delivery instead of the Post Office.
On other sites (for example, this happened when I purchased something from the Metropolitan Museum of Art shop in the USA not long ago), the tax and postage are collected by the seller. You just pay one total sum, tax and postage included, and your parcel arrives, usually by courier.
Most sites will say whether or not they accept international orders, if not up front then in the 'delivery' or 'customer services' section of their website.
I'm sure you know that American sizes of clothes and footwear are not the same as UK or EU ones. So it's a good idea to check the measurements for each overseas company's products. All the websites have size information; it's usually easy to find.