An electron beam can be considered as an intense source of beta- radiation, since beta- is electrons. This is effectively blocked by skin or a couple of mm of aluminium according to A level physics. It's also rather effectively blocked by air, so when you crack open the TV there will be a scary hiss as the vacuum fills.
Highest voltage you're likely to have available will be 400V. I've no idea what happens to a TV if you overvolt it, that's got to be worth a laugh in itself. Electrons accelerated through a single 400V stage aren't going to be moving hard enough to be very exciting.
Mains is AC though, it swings through positive and negative values. Electrons will accelerate towards a positive plate and away from a negative one. Drilling a hole in the middle of the plate & making it positive will give you a beam. Once electrons go through the hole, they are now attracted back towards the plate, so will slow down. If at this point the plate goes -ve, it'll push them away. By stacking a number of plates, decreasing distances apart, you should be able to get them changing polarity at the times required to continually pull electrons towards the next plate, so achieving bursts of high energy electrons.
Not sure what you'd want to point this at though