yea it would good idea

first of all WHO could voice him. I'd say Alan Young.

well, I'll just copy and paste what I wrote on the item page itself:
,, imagine you run around with the demoman, having the Eyelander in your hand, slaying enemies and you get to listen to your dead father which is either proud of you, tells you to be better, greets you to death and taunts you or even says something like ,,come Tavish, now ya'll see the bar of boozening'' as you die. ''

also he could say taunting things and things that are jokes about scotland, booze and movies.
Somehow I got the thought of referencing Evil Dead 2...like the scene where Ash has to fight the head and the body of his girlfriend.

,, Ya fightin' no lass with no noggin', lad. Ya're fightin' men! So dun hol' back! ''

or something like that!

Me scotch just goes right through me!

"Aye That's me boy!"
"A good Demoman wouldn't have blown himself up!" (upon suicide)
"Who brings a sword to a gun fight?" (Apon using no gun in your loadout)
"You know it ain't so bad bein' a floatin' head. Don't got eat, don't gotta use the loo, no arms to blow off..." (Rare continuation (breaking into tears) "No way to hug me own son!"
"I once tried to blow up the queen of England for a Nickle!"
"You have it easy Boy-o. I had to hold twenty-six jobs and I di' not have a mornin' off!"
"KABOOM!" (to be done in unison to Demoman saying the same)
"After this you an me and a bottle of scotch! Though you gotta pay I got no money, cause you know floatin head."

"Tavish, wot the 'ell is goin' on out there? Ah can't see nuthin, remember?"

"Look at those weirdoes and then tell me what they look like."