You need to login on the site, Callofduty.com. Goto theatre mode and link your youtube account to your COD account.
Then you just go into theatre mode, select your video, and an option to render the video will be available and it will process and upload the video to the youtube account.
You actually don't need a Call of Duty Elite account to save your gameplay to Youtube. If you want to save gameplay to your own personal Youtube, then you will need an Elite account. If however you can't link your accounts for whatever reason, simply uploading like normal, and then searching youtube for your gamertag/PSN ID should bring up the video after about 24 hours.
I say 24 hours because the "community channel" usually is processing a ton of videos, so it can take a while for yours to process too. Usually after 24 hours though, it should be processed and you should be able to find it with a simple video search by searching your name.