Use common sense. If the machine cannot be turned on then there is no way to get your information. You have one, and only one option. You must disassemble the iMac and physically remove the hard drive. Then you must find and purchase an enclosure that can power and mount the drive so you can extract data from it. If the drive is defective you would then need to hire a professional data recovery company to attempt to access it. This can cost hundreds of dollars. Are you getting the picture?
If you had a backup strategy while using your iMac then all of your data is preserved in that backup. If you do not have a backup then you are basically out of luck. Before recycling you should physically destroy the hard drive (using a big hammer or very strong magnetic field). That's the only way to prevent someone from seeing your data.