I’d start by saying “No, not like the Muppet,”… except: yes, actually, just like the Muppet. Gonzo the Great is going to be your role model, here. You want to approach this project with cheerfulness, optimism, and an uncritical willingness to improvise reciting the works of Percy Shelley while defusing a highly explosive bomb. This is the point where many people will tell themselves that they cannot, in fact, do that. So this is the first thing that you have to do: admit to yourself that you can do that. You just know that you’ll look like a goofball while doing it. That’s fine. Dignity is overrated.
In fact, that’s the prerequisite for all of this: Dignity is overrated. At least when it comes to yourself as a gamemaster (GM): your players may not want to have an undignified roleplaying game (RPG) experience. Then again, they might, if it looks fun; so make it look fun. Remember, you’re not being a GM to get fame, you’re not doing it to get romantic companionship, you’re certainly not doing it for the money, and I would politely suggest that you not do it for ego-boosting purposes. You’re hopefully being a GM for two, equally important reasons: to tell a story, and to have fun. Basically, do all of the stuff that I recommended earlier here about how to be funny and you’ll be alright when it comes to having the right attitude. Hopefully. I think.
But gonzo is more than being funny: it’s also, at its best, about sheer unalloyed creativity. It’s the kind of interaction that you have with your players where you all come up with a sequence of roleplaying that makes perfect sense (or senselessness) while a particular scene is going on, but collapses into indescribable nonsense after the fact. That level of improvisational role-playing isn’t easy to achieve, but you can always go for the lesser gonzo state* where your players and you really are roleplaying out (and thus, taking seriously) the slightly absurd characters that you’re playing. You’ll know when that happens when your players stop working out the angles before every combat sequence or speech/diplomacy check and instead just instinctively run their characters as if they were those characters.
So if that’s “How,” then what is “When?” Well, I can tell you when not to go gonzo: when you think that you need it to spice up a stale campaign. Unless you’re planning to end it by going gonzo — bearing in mind that doing so will probably mean that the ostensible goal of the campaign will never be reached — it won’t actually fix the problem of a stale campaign, which is (probably) that everybody’s bored with playing that particular game with those particular characters, but nobody wants to say anything because ‘everybody else is having fun.’ Spoiler warning: they’re probably not having fun, either. I’d also recommend against starting a campaign by going gonzo, if only because people will expect that kind of emotional intensity for the rest of the campaign and there are limits to improvisational theatrical stamina, honestly.
Instead, I would recommend waiting until circumstances combine to give you a) a table full of folks who are all in a merry, relaxed mood and b) a natural lull in the campaign narrative. Say, a point where the party’s just gone through the equivalent of Act II in the story, and are now thinking about how to go with Act III. That’s a good place from which to encourage your players to maybe get a little gonzo. Give them some room to indulge in some histrionics, keep the action moving, be very forgiving about the consequences, and hey! They might end up spendomg twenty minutes having a four-sided argument with you, the NPCs, and each other, completely in character, and grinning the entire time.
And even if your attempt to achieve a gonzo state at the table doesn’t work perfectly, they still get to stall for time without feeling guilty about not having a plan for Act III, because after all the GM got them on this tangent in the first place, right? Which is fine: after all, you can always try again later. Even Gonzo the Great couldn’t be Gonzo the Great all the time…
*Fortunately, that state can be achieved through the traditional methods of paying attention to the game, coming up with a meaningful backstory, and staying up to play until 2 AM. There’s a magical spot in the human psyche between “too tired to think straight” and “too tired to keep awake.” Find that spot, and hug it tight, and magic can happen. Or possibly just sleep madness. But sleep madness will do.