- Make sure you know the tax arrangements between your country and the US, and file whatever documentation is needed. In the US, you'll probably want/need to file a form W-8BEN with the IRS (http://www.irs.gov/uac/Form-W-8BEN,-Certificate-of-Foreign-S...) to prevent withholding taxes in the US. Most brokerages will be used to that anyway.
- Documentation of your identity. If you have a US bank account already, this will likely be trivial for you. But sometimes US financial institutions gets finicky about this. I had a US bank account with Chase years ago as I had a small business where I had to process a reasonable number of US cheques, and processing them with a non-US bank would effectively eat everything in fees.
I ended up having to go to the US embassy to get an apostille attached to copies of my passport and the application form (basically someone had to watch me sign the documents, and match them to my ID and then attach a document and seal as evidence that they had certified my signatures). Other times they don't care at all - it appears to be down to who has been subjected to cranky auditors and/or concerns about money laundering recently or not...