Build it into the TV
Whenever you get a new TV, it has a hand-sized panel on one side with two buttons, green "save" button above the panel and red "kill" below (or vice versa) . When you first boot up the TV, you record the palmprint of everyone in the house. The biometric scanner is part of an inbuilt program that tracks which program is being watched and which of the input household members have already voted this session (I'm assuming it's the host who is up for the chop, if theres more than one person to cast a vote about there would have to be a menu popping up to pick which person to vote about).
You want to vote to bump off poor Bill? Scan your hand and push one of the buttons.
Possible pluses for the gov.: If they want, they can build spyware into the TV that automatically inputs all the registered handprints into a vast handprint database. Presto, you have a (nearly complete) database of your whole adult population (and maybe kids too?) Law enforcement (or evil militant secret police, whatever works in this case) can use this to help catch criminals and identify people, and a cross-check of new added handprints to the database can help to prevent folks from gaming the system by registering to multiple devices and voting multiple times. If someone is registered to more than one currently online and activated TV, there are penalties.
Additional thought: to prevent channel-hopping and voting for kicks, perhaps the TV must have been tuned to a program for a certain period of time before voting is unlocked (5 or 10 minutes, maybe)
P.S. I'm just assuming you have some system set up to explain why anyone would be a presenter in the first place (a punishment for pissing off the wrong person, perhaps?) The other main issue I see with your proposal is that people are much more likely to vote if they feel strongly than if they dont care. So those who really want Bill dead are more likely to vote to kill him (because they hate him, think his program sucks, or are just bored, whatever) than everyone else who thinks, "eh, Bill's fine" is to bother voting to save him. Two ways around this are a) keep track of who is watching somehow (camera?) and only have people vote to kill, with doing nothing being a save vote, or b) require everyone watching to cast either a save or kill vote. Enforcing this is probably going to be a sticky problem, though.
P.P.S. Why is being shot with a crossbow humane?