The Lex 35 is a toy/crap camera that's available for US $1.50 from American Science & Surplus. Lest you be tempted to splurge for the Vivitar T100 for US $7.50, be aware that it's the same camera as the Lex 35 except that it has a tripod mount and says Vivitar on the front. Hardly worth the extra six bucks IMO, but these instructions apply to both.
The Lex 35/T100's shutter cocking mechanism is coupled to its film advance, so reliable multiple exposures are out on a stock model. If you've got a screwdriver, a metal coat hanger, a pair of pliers, and some tape, this is not a problem. This will show you how to add a key that will cock the shutter without touching the film advance and allow you to shoot multiple exposures with impunity as well as how to block the shutter from closing, allowing you to keep it open as long as you want.
I say that reliable multiple exposures are out on a stock camera. It is possible to make doubles by pressing up hard on the film rewind button on the bottom of the camera, which [theoretically] disengages the sprocket wheel that advances the film and cocks the shutter normally while leaving the film where it is, but on my two Lex 35s and one T100, the advance always moves. Less than usual, but it won't give me an in-place double. Apparently it works fine on some copies of the camera and not on others. This mod gives me repeatable results without moving the film at all. Your mileage may vary.
The mod doesn't affect normal single-exposure operation, so shoot as usual for that.
Shoot the first exposure on the frame as usual, but don't touch the film advance wheel afterward. Put the key in, use your scratched guide line to line it up, and push the cocking assembly over until you hear it click and remove or back the key out a bit. Once you do, it's cocked and ready to shoot the next exposure on the frame. Repeat as many times as desired. Once you've shot the last exposure you want on that frame, use the film advance wheel as you normally would.
Bulb mode is a bit more of a pain. Put the key in, make sure the shutter is cocked, back it off 6.5mm (you'll probably want to mark that with a permanent marker on the key), and press the shutter release to take the picture. To close the shutter, pull the key out. The shutter tends to close a bit slowly, talking half to a quarter of a second once the key is pulled. Since the key is blocking the cocking mechanism from returning, the key is in control and releasing the shutter button has no effect.
Sorry about the crap lighting. It's a Sunday night and I'm too lazy (and don't currently have the space) to set up a tripod to do it properly, so the glare of the flash will have to do.