This avatar was formerly believed to be randomly awarded by discarding items using the Quick Stock option.In February, TNT confirmed suspicions that the avatar was not random in a Neopian Times editorial:
"So... is the Rubbish avatar truly random when doing a certain task, or are there certain specifications that we must meet in order to get it? It's exhausting me. ~removed Cleaning up Neopia is hard work! However, it's also not randomly rewarded. It may seem that way, due to the somewhat malfunctioning start that it got, but we can assure you it's not random. "
Many theories exist, however it is not commonly believed to be obtainable by discarding 5000 items through quick stock. It is unknown whether there is a time frame, but the value of the objects does not seem to matter.