If you want to cut your neon into shorter strips or splice them together into longer strips, check out this Neon Sign guide by John Park for tips on how to do that.
Flip your screen over so the strings are on the back side, and place the neon right on top. Cut a small piece of craft wire and use it to wire the end of your neon into the screen. Twist the wires on the other side to secure it.
Place a wire at all the corners and any place you feel it needs structure. I found it helpful to use an angled wire to add tension to the ends of the neon, to keep the corners sharper and the shapes more accurate.
When adding shapes, try to lay them out so the wired ends of the neon are as close to each other as possible on the back. This will minimize the amount of cleanup you will need to do later on.
Poke all the neon wires through to the back of your sign. If you've got a soldering iron, put it to use splicing extra lengths of wire between the neon strips. Your wires will show through so be as tidy as you can. It's very possible to hide all the wires behind the neon strips with a little extra wire and effort, if you want it to be perfect.
Use screw terminals at the end of your wires - red to + and black to -. I got mine wired so I only needed one screw terminal, but if you get a handful of them and some power cord splitters, you can use one for each neon strip to make this a no-solder project.
Wire the screw terminal to the screen for strain relief. These wires can pull out pretty easily if they're tugged on too hard.
Plug it in and watch it glow!
If one of your strips isn't working, try reconnecting the screw terminal. These things are notoriously fiddly!
It's easy to take this sign apart and reconfigure it as many times as you would like. Just snip the craft wire attachments and you've got a blank slate to create once more.