The united states does not require citizenship to create a company. You will however need an agent in the us for your corporation, thus a lawyer will be required.
If you are in a country such as the UK this is a simple task, other countries are more complex due mainly to tax treaties, and some countries such as syria are next to impossible due to political embargos.
Typically what you will be looking to form is a us corporation acting as an IBC (itnernational business company). The IBC status with the IRS means that you are not responsible for taxes in the usa. THe united states is actually a tax haeaven in that regards.
Forming a company is super easy, takes 10 minutes and you can do it online for less than $1000, an LLC or a C-corp. Hundreds of places online can help you do it. Generally Delaware, Wyoming or Nevada are considered the best states if your business is not in a specific state.
Getting a bank account is not quite so easy but not difficult. If you go in person to the bank, Chase or several other banks will do it for you. If you want to set up a bank account remotely it will be more difficult.
You don't necessarily need a lawyer, depends on the type of business you want to start. Just to start a company you don't need a lawyer.
The procedure depends on the type of company you are starting, the product you're selling, how you plan to get paid, and who your partners (if any) are in the US.
1/ Form a business entity in one of the 50 states. Among the most popular are Delaware, Nevada, and Florida. This is probably the simplest part since all you need is an address in your jurisdiction of choice and registered agent. The agent does NOT have to be a lawyer.
2/ Open a business bank account. This is probably one of the more grueling processes because you will need to send the bank verification documents. You will also want to set up your payment gateway and merchant account if you're doing sales online -- similar verification procedures apply.
3/ Contracts, contracts, contracts. Make sure you have all your business contracts and documents well-defined, written, and signed, whether they be shareholders agreements or officer/director appointments.
For more resources/guidance, see: http://tinyurl.com/24yb77p