What does “I’m like, c’mon guys. I’m the president of the United States.” imply?

Though the other answers gave a very good implication of hi utterance, I think the OP is trying to parse it too literally and so is not getting the grammatical function of the pieces (so that those pieces can be reused). So for 'I'm like "C'mon, guys, I'm the president" '

• I'm like "...." - introduces something said or thinking. One could change the pronoun. It is mostly synonymous with "I said..." or "I said to myself..." or "My reaction could be described by...". A common way to repeat an interaction between two people, verbal or otherwise, is to say "I was like 'Your momma is ugly', and he was like 'No, she isn't.' and I was like 'Is so.' and then -she- was like 'Sorry, dude, he's right', and he was like 'whoa, that's harsh', and then he was all like making weird faces and then he barfed all over the car seat." That's why you're like 'It smells like some cow died in here'

• C'mon, guys - "Come on" or "C'mon" is an imperative, which could literally mean 'Please follow me...' but here and usually means 'I am exasperated' when others (the 'guys' here) aren't doing something expected. So you could say "C'mon y'all, get off the lawn" or "Come on! The Nixon mask is better than the baby mask for a bank heist". Grammatically, it is saying "(you) come on" similar to "Come off it".

So that's the grammar. The register is pretty informal and colloquial, common in youthful talk. Coming from a former law professor, I'd guess it was meant as a deliberate change in register for effect, a little bit ironically, as though he were some teenager in a movie who was unexpectedly thrown in with the royal family of England and lots of laughs with the difference in speech.

"I'm like" - this can be, as @kiamlaluno said, an informal reporting verb meaning 'say'. However, it can also report unuttered thoughts. Things you wanted to say or were thinking but did not vocalise at the time.

My brother ate the last brownie and I was like "what the hell!" Here, I didn't necessarily confront my brother about his taking the brownie; I was angry/annoyed about it but didn't say anything.

"c'mon" - as you found, this is contraction of 'come on' and here (and often) is used jokingly/mockingly to mean "you can not be serious" or words to that effect.

You're scared? C'mon, it's just a little bunny rabbit. Here, I am surprised that my friend is scared of something so harmless and am lightly mocking.

"I'm the President of the United States" - Here he is saying that his position is important, exclusive, and therefore should carry with it privileges and gravitas. The President of the United States, if anybody, should expect to have cool gadgets and things. It is mock arrogance.

So, the whole quote put together basically means:

I was in the oval office and I realised that it was not very hi-tech. I thought, "Wow! Is this seriously it? I'm the President of the United States. I should have cool gadgets and things."

Hope that helps.

The phrase by itself is a boast, but in this context it's rather a lament.

He means that the public perception is that the President has all the latest gadgets available but it's not that way in reality.

He is justifying his thought of having really cool phones and stuff in his office by claiming 'after all' he is the president of United States.