Below is a list of achievements as reported from within the game's 'achievements' screen - I am not sure if this is a fully comprehensive list of achievements or whether there are additional hidden achievements.
There is a page on the Black Mesa: Source wiki, this has now been updated with information on these achievements.
Pass the toilet paper to the scientist in need.
Perform a surgical caffeine extraction from 20 machines. [0/20]
Kill 10 Airborne Headcrabs using the Shotgun. [0/10]
Kill 20 enemies with the Hivehand. [0/20]
Ruin the microwave casserole.
Kill an Assassin with the 357 Magnum.
Kill 10 enemies with a reprogrammed Sentry. [0/10]
Conduct all of the questionable research experiments. [0/5]
Kill 10 enemies with Snarks. [0/10]
Gray Matter Propulsion
Headshot 20 enemies with the .357 Magnum. [0/20]
Hyper Saturation Conundrum
Drown in Lambda Core's coolant.
Kill a Headcrab using fire.
Kill a Marine with their own grenade.
Kill 5 enemies with one grenade.
Kill an Icthyosaur with either the Tau or Gluon.
Use the alternative path in the Office Complex.
Use an entire clip of Gluon ammo in one continuous shot.
Get overtaken by a Headcrab's amble anterior.
Interfere with the alarm and laptop in Anomalous Materials.
Overcharge it anyway.
Send the Hidden Hat to Xen.
Kill a Barnacle with a Satchel.
Refuse to insert the specimen into the Anti-Mass Spectrometer.
Unified Coupling Theorum
Help the Barnacle and Headcrab find true love. [0/10]
Get killed by your own Snarks.