Beijing is the only place in East Asia with a Turkmenistan consulate. The next closest one is in New Delhi or in the other ‘stans.
For a tourist visa, a LOI from an approved Turkmenistan travel agency is needed. We used Ayan Travel who were able to arrange a LOI in 9 business days (they applied on a Monday and it was sent to us from them the following Thursday evening). NOTE: The Beijing consulate does NOT require the original LOI (i.e. a faxed/emailed copy is OK), unlike Moscow, etc.
Cost for Americans: 10 day single entry is $35 for normal service (one week) or $55 for express (3 day/weekend) service in USD only. They will keep your passport the entire time (so have a copy for yourself as that is against Chinese law). There is a Bank of China down the road on Tian Ze Lu that can exchange RMB for USD. Keep in mind the consulate is closed on Wednesdays (and at 12:00 (at least on the door), not 13:00 as listed on the website). We went with the express option and dropped ours off Friday morning at 11:30am and it was ready Monday at 15:00.
Website: Turkmenistan Visa in Beijing