Nick claims he knows every animal in Zootopia, including Duke Weaselton. How did Nick get to know the weasel and how did he figure that Duke was involved in the crime of turning predators feral?
Nick Wilde claim that 'he knows everybody' in Zootopia was almost assuredly hyperbole, but I would hypothesize that Nick Wilde knew Duke Weaselton because they ran in the same circles. Duke's actions on screen demonstrated petty theft from the Flora & Fauna shop and street corner DVD bootlegging. While Nick's illicit actions were limited to 'grey area' scamming, it was also revealed that he had at least a business relationship with a 'mob boss'. It isn't too much of a stretch to assume that the two (Nick and Duke) knew each other and ran in similar circles. In fact, with Nick's street smarts he likely cultivated relationships with animals like Duke for future endeavors.
But it was Judy Hopps herself that made the connection between Duke Weaselton and the 'Night howlers' (also known as Midnicampum holicithias). After she left the police force and returned home, her father warned some kids (her siblings...?) to stay away from the blue flowers which are toxic. Her father uses then as a natural pesticide but they have the adverse effect of turning a normally docile animal rabid. Gideon Grey is a male red fox who bullied Judy when they were young and he supplies the colloquial 'Night howlers' reference to the blue Midnicampum holicithias flowers.
Light bulb! Judy remembers busting Duke Weaselton with the blue flowers after the theft from the Flora & Fauna shop and subsequent chase through Little Rodentia. She jumps in the family's truck, makes her way back to Zootopoia where she locates Nick through Finnick, a male fennec fox and sometime accomplice.
Surprize! Nick knows Duke Weaselton and where he can likely be found (selling bootleg DVDs on a street corner complete with big red banner and card table). They get the clue to the next part of solving the conspiracy from Duke.