Canonically, we know he was the previous Royal Scientist.
We know he built the Determination Extractor, or at least designed it.
We know he built the CORE.
We know he was shattered across time and space.
We know he made the Gaster Blasters (since they're named after him), and that Sans and (most likely) Papyrus have them, which hints at their connection.
We know that his full name is WingDings Gaster, and since Toby Fox confirmed that the skelebros have another relative, he is most likely related to them as his name is also a font.
We know he speaks in Wingdings and that he wrote Entry Number 17.
We know that Gaster's Theme, accessed via the Sound Test Room, is present in both STMPWYFS and Nyeh Heh Heh, further hinting at a connection between him and the skelebros.
We know that both Sans and one of the Gaster Followers mention that 'it's rude to talk about people who are listening', which hints at- wait for it- MORE CONNECTIONS.
Anything else?