The simple answer is, that it varies. Here's what I'd consider:
The communication medium
If you're using Skype (assuming the instant message component, rather than video chat) there's less expectation to be formal. What you type in an instant messaging app should be conversational, and because you're communicating in real time, it's acceptable to abbreviate for the sake of brevity.
If you're using email, expectations may differ, so read the other considerations.
If you're writing an actual letter to be sent via physical mail, it should be formal. Common abbreviations like ASAP should be avoided, although you can declare a cumbersome proper noun as an acronym and reuse it throughout. You can do this by writing out the name, followed by the acronym in parentheses - for example "Federal Bureau of Investigation (FBI)".
You should also consider the ubiquity of your acronyms and abbreviations. ASAP and FYI are common, and you'd expect anyone to know them. Acronyms like SDLC or CMDB are common parlance in Information/Communications Technology, but would likely draw blank stares from anyone else. Younger people are more likely to use and recognise a broad range of communicative abbreviations.
I'd also tend toward formality based on the hierarchy in your organisation - I'd write formally to a CEO/CFO/CIO, or even my immediate supervisor, but be more casual with my peers.
How formal your communication should be also depend on how well you know the person you're communicating with. I'd try to be formal with anyone you're unfamiliar with, and adapt your style as you get to know them. If it's someone you've never communicated with in any way, your communication style is your first impression, and from my experience people are less likely to be offended by an overly formal approach.
Over time, you'll get a sense of what they're comfortable with and respond in kind, just like you would with verbal communications.
As a caveat to this - you should consider more than just your immediate audience. For example, if you're in customer service and your chat log is auditable (as a measure of your own performance, or as part the record of service provided to the customer) then your audience could feasible be anyone, and you should avoid being overly familiar.