The character most associated with Dave Goelz is none other than The Great Gonzo. In some ways, studying the growth of Gonzo's character is also a look into the development of the skills of his puppeteer.
Before "The Muppet Show" Gonzo was just a Frackle called Snarl who made his debut on a TV special called "The Great Santa Claus Switch." The puppet was handed over to Goelz as "The Muppet Show" began production.
During the show's first season, Gonzo was kind of a sniveling, pathetic looking guy with a nasal voice. He was more of a strange performance artist at this point, specializing in acts such as eating a tire to the music of "Flight of the Bumblebee" and demolishing a vintage automobile with a sledgehammer to the music of "The Anvil Chorus" (followed by eating the crank shaft acapella). He was an interesting character, but it wasn't until Season 2 that he really began to shine.
Between seasons 1 and 2, Goelz, who began his career with Henson as a puppet builder, redesigned the Gonzo puppet. The shape of the nose was changed a bit, and, most notably, the eyes were made larger and a surprise mechanism was added. Whereas before the eyes cold only look droopy, now they could look excited, which I think was a big key for advancing the character. A study of the changes to Gonzo's features, "Gonzo through the Years," can be found at the Muppet Wiki.
With a more expressive face, Gonzo's body language began to change as well. In season 1 Gonzo was kind of stiff, but in season 2 we began to see a calm Gonzo and an excited Gonzo. Calm Gonzo is somewhat like he was in Season 1. The eyes are droopy, head pointed down somewhat, and the arms are held in close to the body. The excited Gonzo is more...well...wide. The eyes become wider, the mouth opens wider when speaking, and the arms spread out wider. What's great is watching the transition between the two. A fantastic early example occurs in the episode hosted by Rich Little when Gonzo asks Kermit if anyone has showed up to audition for his new dancing chicken act (By the way, this is the first appearance of Gonzo's fascination with poultry). It's calm Gonzo who enters the scene, but as he starts to describe the idea for his new act the eyes open up, the arms get wide, and the voice pitches up. He's excited about this great new act and it shows. This all happens gradually over the course of two sentences.
A few minutes later we see the same thing happen again, only much quicker and to a greater extreme. When Kermit turns down his request to have the dancing chicken on the show, Gonzo goes from calm to excited x10 in just the space between Kermit's line and Gonzo's response.
"But Kermit, why!?!"
This time, Gonzo gets himself so worked up that he's left panting as Kermit responds to. Breathing is such a subtle thing to add to the manipulation of a puppet, but here it helps Gonzo to look alive and to help sell his emotions.