I can say I've skyped someone. But when referring to chat, such as ICQ, is it correct to say that I've chat someone, with the meaning that I've sent someone a message on chat?
What you are talking about is called "Conversion" in linguistics (turning one form of a word into another) and more specifically "Verbification" in the case you are asking about (changing non-verbs into verbs).
So specifically answering your question —
Yes, you can Skype someone, but you won't likely hear "chat someone" in correct usage. Even "Skype someone" is somewhat new and would likely only be considered acceptable usage in informal circles. That is simply by convention simply because one came into common usage, while the other… hasn't.
Honestly, there are no rules that determine when a noun can become a verb. It is simply a matter of when it becomes common enough usage to become acceptable.
You can "phone, fax, or text me" but you can't "cell, smartphone, or iPhone me" — Strange right? Wait, it gets crazier — You can "bus someone" across town, but you wouldn't "plane someone" across the country. You can "gun someone down" but you would not "pistol someone down." You can "hammer a nail" but you wouldn't "screwdriver a screw." You can "milk a cow" but you can't "egg a chicken."
Fortunately, you can search the web to determine if something has become common usage, but be sure to use canonical or at least well-vetted sources (newspapers, books, articles, etc), not chat, street talk, or forums… which are more prone to slang and unconventional language use.
In my understanding, when you say "I've skyped someone" you are referring to the act of "calling someone", which then is understood as speaking with someone.
Chatting would be the act of speaking itself, so both are not comparable.
Now, that being said, "I've skyped someone" is incorrect, but understood, as "I googled this or that" is. "I chat someone" is both incorrect and senseless.
One wonder of the English language is that any word can be verbed.
You can say:
I [have] chatted [someone].
It's very colloquial, but will be understood. More correct would be:
I chatted with [someone].
They have slightly different connotations.
I have chatted Bob.
This suggests that you sent a chat request to Bob; he may or may not have responded, and a conversation ensued.
I have chatted with Bob.
This suggests that you and Bob engaged in a mutual chat conversation.
Another use of the word chat would be:
I chatted up Bob.
This would mean you spent a long time chatting to Bob, likely for the purpose of obtaining favor. Many people peruse bars to "chat up the girls."
As others have mentioned, English does have a great tendency to change words into whatever part of speech is required, including verbification. Thus, the brand names Skype and Google easily give rise to the verbs skyping and googling, respectively.
However, this tendency has limits. Namely, the language tends to resist changing a word into something that already exists. Successful coinages are successful because they fill a void; if there is no void, there's no point inventing a new word for it. Chat is already a verb, and has been one since long before computers were invented. Therefore, there is no need to invent an awkward transitive "I chatted him" when there is already the well-established intransitive "I chatted with him".
That said, not having a point doesn't always prevent people from inventing words anyway. In a sense, googling is a pointless coinage, because there was already the perfectly good verb searching. There's no rules or even predictability about this: some coinages take off, others inexplicably don't. (And in fact you will sometimes find people using "chat" with a direct object, sometimes even with a different shade of meaning than it would have without a direct object. It's just that these usages have an extra "that sounds wrong" obstacle to overcome.)
All you can do, especially as a learner, is to follow established practice as much as possible: if you hear multiple native speakers use a particular expression, it's probably fine to imitate them, but it's generally best if you don't go around inventing usages of your own, at least not until you're very, very comfortable with the language.