How can I change my username to scrambled letters? Is it even possible?

You can't.

The way this is done is using formatting characters not allowed in usernames, so it is impossible.

You can make anything in Minecraft scrambled letters (called "obfuscated" in the code) by writing §k (U+00A7 k) before it. However, for this very reason, the § character is disallowed in entry in most things in the Minecraft GUI, requiring you to insert it with a level editor or external tool—and in regards to your particular use here, non-alphanumeric characters such as § are not allowed in Minecraft usernames (only ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789_ are allowed).

However, if you have a third-party launcher, configuration edit, or mod that lets you launch with an arbitrary name, and if you join a server that is offline-mode/cracked (i.e. doesn't check that your name really belongs to a Minecraft account you logged in with), you can join that particular server with a §k name if it doesn't check explicitly for it. I have done that before.