There are a few ways.
1) (Both Corporate/Personal) : Open a bank account locally with a bank that also has branches in the USA - then use their international banking center to facilitate the process of necessary documents at your local bank (I.e. HSBC, Citibank offer this service) - http://www.citibank.com/ipb-global/homepage/newsite/content/... , http://www.expat.hsbc.com/1/2/
2) Just personal: Open an "High Net Worth" account. Also can be done from anywhere in the world, but you must usually keep at least $30K USD in the account liquid in order not to be charged hefty fees. Citibank offers these accounts as well (see link above)