When Harry wakes up after his nasty nightmare in which Voldemort murders Frank Bryce, he wishes he could talk to "someone like – someone like a parent: an adult wizard whose advice he could ask without feeling stupid, someone who cared about him" (2.28). Suddenly he thinks: Sirius. Harry looks up to Sirius Black as a father figure in Goblet of Fire (which sets up some problems for Order of the Phoenix, but we're not there yet).
In Goblet of Fire, Sirius operates mostly offstage. Because Sirius is still on the run from wizarding law (having been falsely convicted of killing Peter Pettigrew thirteen years prior), he can't exactly chat with Harry out in the open. He and Harry exchange letters often, though. Harry talks about his bad dreams and the Triwizard Tournament and Sirius warns Harry to be careful. But as Harry's experiences with the Triwizard Tournament grow more and more dangerous, Sirius makes his own dangerous decision to return to England and to hide out in a cave near Hogsmeade (the nearest wizarding town to Hogwarts).
Luckily, Sirius can take the form of a large black dog too, so it's pretty easy for him to stay unrecognized. Even so, it's a stressful, hungry life he's leading to be close to Harry. As Ron comments, "He must really like you, Harry [...] Imagine having to live off rats" (27.211).
Sirius brings Harry a great deal of background information: he knew Bertha Jorkins, and he can't believe she would just disappear into thin air. He's also deeply suspicious of Mr. Crouch. He tells Harry that Barty Crouch, Sr. was an incredibly ambitious wizard who conducted a crusade against Dark Arts users back in the day. Thanks to Barty Crouch, Sr., Sirius was thrown into Azkaban Prison without a trial and had to rot there for twelve years. Sirius also doesn't trust Snape at all, and he wants Harry to steer clear of Karkaroff, the headmaster of Durmstrang (and former Death Eater) as well.
Sirius's role is primarily as a confidante and adviser in Goblet of Fire. He doesn't do much directly, though he's there to sleep by Harry's bedside in dog form once Harry is brought back from the third Triwizard task. His behavior in this book, however, is clearly setting up plot arcs that will be carried on in Harry Potter and the Order of the Phoenix. Sirius's eagerness to risk his own health and safety to help Harry is one theme that keeps returning.
Another lingering problem is his absolutely unshakable hatred of Professor Snape. Dumbledore makes the two men shake hands at the end of the book, since they are "on the same side now" (36.179). Even so, they continue to glare at one another "as though they wished the other nothing but ill" (36.179) – after all decades of hatred doesn't disappear in one moment. Since this happens in the second-to-last chapter of Goblet of Fire, it's clearly going to be important in the next book.