The plastic I would like to use is Polycarbonate, which will melt if you try to solder metal to it. If you can't weld the two, is there another way in which you can join metal to plastic? Ideally, I would like to build the circuit board into the polycarbonate itself.
You should consider building your circuit on a thin (flexible) polyimide substrate, and then either gluing or heat-staking that circuit to your polycarbonate case.
Flex circuits are as easy to design as regular rigid PCBs (although there are a few more design rules), and there are many manufacturers that can produce them for you.
Two options. One easier to reproduce than the other.
The first is inlaying, which comes in two forms of application. First you mill or cut out a channel. You can use a standard bit or an inlay or keyhole type bit to privde better grasp. Then you can 1, cold form a conductor into the channel (read: hammer it in) or 2, hot form it (pour molten metal). Heck, get a lot of low temp solder (lower than the burning/melting point of your plastic, and a heat gun.
The other is copper tape. You just cut and tape it to the plastic. If you want to get fancy, you use copper sheets in a mill/plotter. You can get them with glue umm, glued on, or without, and use your own spray adhesive.
Either can be done by hand or machine.
Depending on your needs, conductive glue or ink might work as well. It really depends on how complex your circuit is. Invest in some kapton tape (high heat resistance) to protect the plastic while soldering on it, use the least wattage iron needed, and practice your ninja soldering.
I know this topic is 5 years old now but I want to share something. My keyboard was spilled some juice and CTRL and Caps Lock keys were not working. I tried to fix it but some traces on the plastic shield was burnt therefore blocking the electronic contact between the main PCB and the keys.
What I did think to do was creating a bridge over burnt traces to get the contact back. I tried directly soldering it but since plastic doesn't really matches with solder, it didn't catch it and I wasted some of solder during that process. I even tried silicon but it didn't work as expected.
So while I was searching something useful in my tool box, I saw staplers and that crazy idea popped in my head! I can use this stapler pins to create electronic contact so I can solder the cable which I'm going to use it for the bridge directly to the stapler pin! I stapled one across the health trace and other one near CTRL and CAPS Lock traces. I solder a normal thin wire to these two stapler pins and voila, it came back to life.
If you are having the same problem, feel free to comment down. I'll see it in less than 24 hours since I'm actively searching for answers on stackoverflow!