I bought Destiny on the Xbox 360 Version a while back, Now I have the Xbox One and I don't wanna re buy the game. Is it possible I don't have to buy it again
While Destiny is not compatible with the backwards compatibility listed in Slayner's answer, it is possible that you are entitle to an upgrade, if you are a "Year One" player.
Bungie ran a promotion where "Players who purchased Destiny before or on the 15th of January, 2015" were given a free upgrade to the next generation consoles format. This was to acommodate for the fact that players would not wish to purchase the game on both generations of console, and many had not yet made the upgrade.
This applied to Destiny, The Dark Below, and The House of Wolves. Having only played on next-gen, I can not confirm how this worked; however, when you log in to the Xbox 360 copy, you are playing the same account as you were when you logged in to the PlayStation 3 copy. It is likely that this "upgrade to Xbox One copy" has automatically applied to your account. This upgrade does not apply to The Taken King, and will not apply to any future content, unless directly advertised in future promotions.
Another user is having a similar issue, but it appears they were expecting The Taken King to be available, due to the availability of Destiny without the intentional activation. It might be different, again, on Xbox.
After going to the xbox.com web site :
-If you have bought it online it will be link to your account and could be donwloaded for free.
-If you have bought it offline (box +CD) you just need to put it in your XboxOne like a normal game and it should be able to automaticaly donwload it and instal it to your Xboxone.
If it doesn't you need to contact the XboxOne support team to see with them why it did not work.
Apparently not all Game are considered as compatible with the XboxOne (you can see the list on the Xbox Web Site).