In my opinion, what you ask isn’t possible the way you describe it. Yes, it’s possible to become a best seller, you can become highly sought after and you can get customers to prefer you over others. In no time, though? I don’t think so.
Even to become moderately successful on Fiverr it takes effort, some struggle and especially time. To become highly successful is like running any other business. You build a reputation. How does one build that overnight? There have been a few cases of “luck” here in a sense. If someone comes up with a truly unique gig that catches on, it can go viral like anything else online.
The more gigs that go up every day, though, the harder is is to find an idea like that. Even if you did, sustaining the success would still take a lot of work. This isn’t to be discouraging at all. Fiverr can work. I just think it’s unrealistic to think that there is some magic bullet.
Edit: If there is one thing that could accelerate your chances it would be investing in advertising after at least level 1 and more likely after level 2. If you are asking for the closest thing to speedy success, you’d have to hire someone with real advertising experience and let them handle your campaign. That’s not cheap though, and there is no guarantee whatsoever, especially if your gigs aren’t as good as you hoped.