The sink needs to drain into your wastewater or it will quickly overflow, but it doesn't need pressurized water coming from the faucet.
Here's how I hooked up mine - have made some changes since that video, but that's the general idea:
http://www.youtube.com/watch?v=uWTX5CqEALw
EDIT: sorry dude, I didn't quite realize what you were asking - the answer is "yes", you can mount the solenoid under the sink, but why would you want to? Either way you're going to have to connect the water output to the input of the Hydrogen, so if you're thinking that hooking up the solenoid would be a more "discreet" way of doing things, I'm telling you that it wouldn't.
To answer your question(s), as TF said you're gonna need a "T" off of the feed hose from under the sink, and that will allow the sink to continue to work.
If you're not worried about the sink working (although I'd recommend keeping a working sink intact - I'd suck a dick for a sink in my grow space), then you can simply insert a 90 degree corner under the sink and bypass the faucet completely.
Still not sure what benefit this would give you - are you worried about the connection between the faucet and hose? They make adapters for that (if you are).