Here's my analsis of his personality;
When Scooter first appeared he assumed he'd be welcomed with open arms to increasing the Muppet Theater staff. He was polite (if a bit demanding) when he said he was going to be the go-fer, then seemed like he was whining when Kermit refused. Finally he looks away mentioning his uncle. That's where you can tell he's not such a bad guy. He seems a little embarassed and ashamed that he has to use his uncle as a trump card, though sees that that was what got him the job, he uses his uncle again to get $30 a week, the other times he's probably defending his uncle from George's insults and just mentions one act is his favorite in "Jim Nabors".
His relationship with Kermit is a growing one. At first he sees Kermit as the easily-intimidated show host who's scared of being out of a job if Scooter is mistreated and uses it to get what he wants, though "Simon Smith and his Amazing Dancing Dog/Bear", the Wind-up TV show host, and Gonzo's female impersonation attempt seem to be as far as he gets before Kermit starts to show some backbone towards him. Maybe this is what makes Scooter start to respect him more and not behave like the brat he originally was, as in "Lene Horne", he repeats what Kermit says about being "Merciless and cruel" and tries it out on Piggy, to his misfortune learning she doesn't care who his uncle is. After that he refers to Kermit as "Frog Honsho o' mine", "boss" and "Cheif", showing he has much more respect for him.
Perhaps it is that reason why he actually seems to take his job as go-fer more seriously later than in earlier episodes.
His relationship with Fozzie is a little odd. He may or may not be just trying to boost the bears' ego when he says he thinks Fozzie's funny, since he doesn't get the joke Fozzie then does for him, despite the great joke Scooter gave him earlier, leaving us to think Scooter is either really the "Amusing little twit" Piggy calls him later or has a good sense of humor but poor timing or comprehension of said joke.
He does seem to think of Gonzo as a friend, as when Fozzie listens to him through his door, you can hear Scooter calling "Hey, Gonz". He also takes it on himself to be the little Frackle's manager, insisting Gonzo needs help, now who but a friend would do that? These two are also seen chatting it up on how the crowd actually loved one of Gonzo's acts for once while Fozzie is trapped in Marvo's box in "Ben Vereen". Maybe it's because Gonzo was a bit like Scooter himself, a little bratty in always wanting things his own way.
With Piggy, he has learned to do whatever she wants not out of respect or a longing to help, but of fear of being hurt.