Q:

What does the author want to convey by ending his letter with »Tschüssikowsky«?

The article “Ich hab Urlaub - und Sie nicht” from Der Spiegel provides examples of Christmas out-of-office autoresponder email messages. One example, illustrating a potential ending to an email message, reads:

Tschüssikowsky, Klaus Kalau

Tschüssikowsky is obviously a humorous way of saying Tschüss. But what exactly does the author want to emphasize by using Tschüssikowsky? (There's no context; just the above example given.)

A:

He is just trying to be funny. Tschüss is mostly used in the northern part of Germany. Tschüssikowsky is a stultification.
According to Wiktionary, this term was first used in the German translation from The Persuaders! (Die Zwei, 1972) 

A:

"Tschüssikowski" is a somehow humorous version derived from "Tschüss" which simply means "Good bye".
I don't know if there's an official name for transforming a common "standard German" word or phrase into something theoretically senseless that however sounds similar enough to the original to be recognized, but that is what has been done here. Sometimes it is done by mixing unrelated words together or appending unrelated endings, sometimes there are just some twisted letters, sometimes an entire word from a different topic is used instead of a similarly sounding one ("Malapropism").
Such humorous phrases were somehow quite popular in the past (before my time, maybe in the 80s), but is considered pretty annoying by most people nowadays. From my perception it's no longer really commonly used, except maybe by a few people who conserved that kind of humour.
Some more similar examples that belong to the same category IMHO:

zum Bleistift (zum Beispiel)
bis Baldrian (bis bald)
Wirsing! ([Auf] Wiedersehen!)
Schlepptop (Laptop)
teflonieren (telefonieren)
Schankedön (Dankeschön)

and so on...

A:

As a German, I think it is mostly used to say: "Bye, I'm out", but in a more humorous content. Additionally it may also transfer, that this is his or her (hopefully, but not in a negative way) "last word". 

