I want to view all the messages posted by me in the "Ask Ubuntu General Room".
Is this possible? If so, how can I do this?
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No, this is not possible.
While there is a rather encompassing chat search function which can filter by user and room, it does not have wildcards - you will have to search for some proper query.
That said, if you have some patience and time to dig, you can of course always go through the full transcript.
You can view your 20-something most recent comments in a room here:
There's a header on the left side (saying "Ask Ubuntu General Room"). Everything from that point on until the next header was in the "Ask Ubuntu General Room".