Depends a bit on where you live and when it was purchased; I'll assume the new WEE directive doesn't apply in this case (Europe; only recently introduced; means manufacturers/retailers are legally obliged to take back old electronic waste).
Most recycling centres/domestic waste places have a special dump for old electronic equipment. They are broken down and heavy metals etc. removed/recycled.
If you have access to crushing/grinding equipment, the best thing to do is grind it down into little bits and scatter them over pristine wildlife habitat. Animals are usually desperately in search of food, and plastic and metal contain valuable nutrients and minerals.
Alternately, just chuck the player out there, along with a few of your movies. Because the animals do get bored.
When my first DVD player broke I sold it on eBay with the intent that maybe someone out there could use the parts. I clearly stated in my ad that the player did not work, and low and behold, someone still bought it for about $20. Just an option....