How can I get an existing DA:O character image uploaded to the BioWare site?
There are several possible problems and related solutions, but in general, this issue may not have been addressed as there were a lot of defects in that system.
An issue of time
Wait. You should note that it takes a long time for the bioware social site to update as well and that EA has also been having server issues which might also be related. This feature has been rather hit or miss pretty much since launch. You could try repeatedly to trigger an upload again, but with no way to guarantee you'll reach their servers, it all comes down to luck.
Upload settings
You need to make sure that your upload settings in DA:O are correct and that you are signing in with the same account on your 360 that you are using on the Bioware community site.
360 specific problem
Repeated attempts and only upon character creation was the only solution for this specific problem. There was an issue with uploading character portraits and data on the 360 that had been there since launch and the thread never stated the problem ever being resolved. The important post in the linked forum thread states
...character portraits only upload upon character creation. If, for whatever reason, you are not connected to the DA servers when you make your character then they will NEVER upload their portrait to the servers or this forum.
Worse still, even if they did upload, they did not keep up to date and were always stuck with the starting armour.
There is documentation of this happening even a year ago with no fix as well but in that case, one poster blamed the extreme latency with that particular feature. As someone thought that to be the explanation, it would seem to have worked for some, if slowly.
Other issues and solutions
There was an issue with bioware's online storage filling up and not updating, etc. You may want to check that you have space to upload to. This was more an issue with the PC version and involved screenshots filling the storage and character data not being overwritten, leading multiple versions of the character data files filling up the space so I doubt it is related, but it is a possible explanation.
If it's more about the character's basic features in the portrait than anything, you could use the Character Creator on PC and create your character as that will most likely work to get a basic portrait up. Correct me if I'm wrong, but I'm pretty sure you won't be able to use the character on 360 or change the armour or anything.
If all else fails, you may try contacting EA support for more assistance. Their live chat is decent.