Use the Xbox netowork Diagnostic tool to test your connection. I'd suggest doing it multiple times to make sure you get a good idea of what your stats are. Destiny doesn't need much for up/down speeds. 1up/1down should be ok. Your latency needs to be low. under 50 is best, under 100 is ok, under 150 will work but you may notice some lag, over 150 you need to work on getting it lower to have a decent experience/not lag out. If you have any packet loss you will have problems. Try unplugging your router, modem and xbox for 5 minutes if you do have problems. Go through the network guide and make sure your router is properly configured. IF you happen to be on wireless, get hardwired.

Have you tried power cycling your home networking equipment, router/modem, by unplugging them for at least five minutes?