Based on the song "Like A Boss" by Andy Samberg and Lonely Island Boys.
The lyrics are about the responsibility of power or lack thereof. The concept behind the song is loosely based on individuals in contemporary pop culture (i.e. rap stars) who insist they're the most amazing person in the universe but are, in reality, rather ridiculous.
This scenario directly effects the meaning of the phrase. While most would say it signifies an increased level of "awesomeness" in a given situation, the context the phrase was originally used in would suggest the opposite.
The point of it is that you're saying you're awesome using a terminology that's based on making fun of people who say they're awesome. The irony makes it hilarious, especially because the term has run rampant and most people who say it don't realize that they're, in essence, making fun of themselves unknowingly. Double funny.
It can also be used in a sarcastic, summarizing capacity (see example below).
With the irony of it in mind, you can interchange it with longer explanations as I did in a Facebook post once:
By saying, "like a boss" what I'm really saying is, "my very busy and overextended work schedule is contributing to a substantial increase in the amount of alcohol I consume and it's not only pathetic, but also not exemplary of the type of behavior that's indicative of upper management."
Just realized it's 6:12 PM and I was SO busy today that I didn't eat, I still haven't checked my voice mail and I've had to pee for the last 4 hours. Who needs a glass of wine? This girl right here. Like a boss.
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