While I agree with the previous posters about using MP3Tag or Jaikoz, I still want to give you an option on how to do this with Picard, since you asked in the Picard section. The reason other tools were reported first is probably because of Picard really being made for tagging with MusicBrainz data, and as I understand it you just want to change the existing tags on your files and move the artist name into the album tag. But you can edit local tags with Picard as well, and you actually can do what you want pretty easily with the help of a little script.
First create a new script under Options > Scripting, give it some descriptive name (e.g. “Set artist to YouTube”) and set the following script:
Make sure to not activate the script (keep the checkbox in front of the script’s name disabled), since you probably don’t want this to be executed for all files. Then load the files you want to tag this way into Picard, select them all, right click and choose “Run scripts…” > “Set artist to YouTube” (or whatever you named the script). This will change the tags, you then just have to save the files.