First i would have to ask if you can turn on you MacBook while is pulled in, if so the charger itself is fine. The problem would be in the battery. If it will not power up while pulled in you would need to test it with a volt meter to see if it supplying power or not. If it is supplying power then the problem lies in the computer and there are many guides to help you repair it.
There is a simple solution that will work until you can afford to buy a new computer. On the back of your iMac there is a mini-DVI output port with support for DVI, VGA, S-video, and composite video connections via adapter. This will allow you to connect to a different monitor or even a newer TV.
This iMac supports a maximum of 16GB of 1333MHz DDR3 SDRAM. Depending on your budget you can for the (4) 4GB sticks at now or buy (2) 4GB sticks now and a additional 2 later. note: this speed of memory is every common right now so DON'T trow away the (2) 2GB sticks, you may need them one day.