How does Clarice know "Jack Gordon" was actually the real Buffalo Bill?

Your question seems to be more about how quickly Agent Starling finds 'Jack Gordon' in the final scenes of the movie - rather than how she was able to conclusively determine that he was Buffalo Bill (i.e. the moth and then when he attacks her).

If I recall correctly there were several clues that led her quickly to his house, and from there to determine he was the killer. Its been a while since I've seen the movie, so some of this may be on shaky ground - please don't hesitate to correct it:

1. The most important clue comes from Hannibal Lector - he surmises that as the murderer is clearly harvesting skin to make himself a suit, and Lector's psychiatric training and homicidal nature enables him to paint a good profile of the likely murderer. However he only releases this information slowly to Clarice in return for personal details about her, and the clues are often in the form of riddles or his own twisted jokes. When talking with her towards the end of the movie, he offers a hidden final clue to her ... 'we covet what we see every day'.

2. I seem to recall that Clarice looks out of the window of the first victim's bedroom - perhaps hypothesizing that Bill saw the victim from the very room she was in. It seems reasonable to assume she prioritized her house-to-house inquiries based on the view out of the window. Clarice had only just found some semi-nude photos of the victim hidden in the room, perhaps hinting that the victim had an exhibitionist side to her - perhaps she didn't mind being seen from her room (this last bit is somewhat conjecture, but I don't see why those photo's are important otherwise)

3. Jack Gordon shares initials with Jame Gumb and John Grant - perhaps all are aliases. He gives his name early in the conversation with Clarice - surely this would alert her and make her want to stay talking to him.

4. The moth is the final indisputable clue - it is a species that is not native to the USA, and only a few people rear them.

It wasn't Mrs. Littman that Buffalo Bill asks is fat, it was the Bimmel girl. Also, what led Starling to the house was the fact that it was believed to be owned by Mrs. Littman, who the Bimmel girl had previously worked for. This information was given to Starling by Bimmel's friend. Upon arriving at the house, Starling encounters Jack Gordon (aka Jame Gumb), who indicates that Mrs. Littman had died and he is the new owner. This is how Starling finds the residence.

What is puzzling, however, is that Bimmel's friend did not appear to know that Mrs. Littman was dead. Considering that Belvediere gives the image of a small town, this would seem to be an oversight. Not having read the novel and judging from this information, would could infer that Buffalo Bill killed Mrs. Littman and took over her house. In fact, I believe the body Starling finds in the bathtub is possibly Mrs. Littman. Therefore, Starling accidentally discovers Buffalo Bill, who may have been assuming Mrs. Littmans bills and other matters surreptitiously. This would also explain why his residence in Calumet City was vacant.

In addition to the moth (which ultimately seals the killer's fate), Gumb acts very nervous, which raised concerns with Starling. In addition, there is one shot of the house while Gumb is looking for the phone number for Littman's "son." Some of the items (such as the shrink wrap) and the disheveled state of the house would have caused some alarm with Starling.

This is answered in the IMDb FAQ:

How does Clarice know that "Jack Gordon" is actually Buffalo Bill?

Her suspicions are aroused when "Gordon" asks if Mrs. Littman was "a great big fat person." Lecter pointed out that all of the victims were larger women, because the killer wants their skins to fit onto a man's body. That Gumb's first thought of a woman he didn't otherwise recognize was her size was enough to make Starling want to find out more. Once inside his house, Clarice recognizes the Death's Head moth in the living room. Gumb gives his name as "Jack Gordon." Those initials, J.G., are consistent with "Jame Gumb" and "John Grant," both aliases named by Crawford.

Starling and her friend, Ardelia Mapp, have a conversation (after the scene where Lecter escapes from custody). Staling notes that Lecter said that "everything we need to catch him is right here in these pages". The subsequent dialogue goes like this:

Ardelia Mapp: Is this Lecter's handwriting? "Clarice, doesn't this random scattering of sites seem desperately random - like the elaborations of a bad liar? Ta, Hannibal Lecter."
Clarice Starling: "Desperately random." What does he mean?
Ardelia Mapp: Not random at all, maybe. Like there's some pattern here...?
Clarice Starling: But there is no pattern or the computers would've nailed it. They're even found in random order.
Ardelia Mapp: Random because of the one girl. The one he weighted down.
Clarice Starling: Oh, Fredrica Bimmel, from... Belvedere, Ohio. First girl taken, third body found. Why?
Ardelia Mapp: 'Cause she didn't drift. He weighted her down.
Clarice Starling: What did Lecter say about...?First principles"?
Ardelia Mapp: Simplicity...
Clarice Starling: What does this guy do, he "covets". How do we first start to covet?
Ardelia Mapp: "We covet what we see -"
Clarice Starling: " - every day."
Ardelia Mapp: Hot damn, Clarice.
Clarice Starling: He knew her.

This is how Starling knows that Bimmel is the key and that the killer is very likely a local. She proceeds to Belvedere, Ohio to Bimmel's home where she investigates the girl's room.

Another IMDb FAQ concisely outlines the ensuing events:

Following Lecter's advice -- that Bill covets that which he sees everyday -- Clarice drives to Belvedere, Ohio to talk with the parents and friends of the first victim, Frederika Bimmel. In looking over Frederika's bedroom, Clarice notices the the triangular gussets on Frederika's dresses match the pieces of skin Bill cut from one of his victims. Clarice calls Jack Crawford (Scott Glenn) to inform him that Bill is making a "woman suit" of real skin. Crawford is already in route to Chicago, having identified Bill as Jame Gumb, but they find that the house where he's supposedly living has been abandoned. Clarice continues to poke around in Belvedere and is eventually led to the house of "old Mrs Littman." Mrs Littman doesn't live there any more, but the current resident, who calls himself Jack Gordon, offers to give her Mrs Littman's son's business card and invites Clarice inside to wait. When she notices spools of thread and a Death's-Head moth, she realizes that Jack Gordon is really Jame "Buffalo Bill" Gumb. She pulls her gun on him, but Gumb leads her on a wild goose chase into the basement, where he has Catherine Martin imprisoned in a well. Gumb dons a night vision mask and turns out the lights, leaving Clarice to feel around in the dark. When he makes the mistake of cocking his gun, Clarice shoots him. Catherine is subsequently rescued. In the final scenes, Clarice is shown graduating from the Academy. At the party that follows, she receives a phone call from Lecter, calling from the airport on the island of Bimini in the Bahamas. He assures her that he's not about to come looking for her and requests that she extend him the same courtesy, which Clarice cannot do, of course. He then says that he must hang up because "I'm having an old friend for dinner," as he watches Dr. Frederick Chilton deplane.

_{^{While the characters in the movie might have been saying Mrs. Littman, the subtitles and the book both use Mrs. Lippman.}}

To summarise further, when Starling gets to the killer's house, she already knows a few things about him besides his psychological profile:

• He is a tailor/dressmaker.
• He has a thing for butterflies/moths and is known to have raised a Death's-head moth.
• He has a thing for "great big fat" people.
• He uses aliases using the same initials (J and G).

Each of these checkboxes are ticked during her meeting with the killer leading to her epiphany.

It is never revealed, but in the discussion between Starling and Lectur after he meets the Senator, he asks her questions to get her to put together the reason Buffalo Bill is killing.

Now think on the exchange. He repeats the word Simplicity.

Simplicity was or still is a company that made patterns for women to sew at home.

Now she doesn't pick up on this, and it would be a leap for this to come back as a way for her to put this together. it is more a foreshadow of a sort.

But, Starling sees the dress on the hook on the closet door. It has areas where white material is sewn on which reminds her of Bimmel's wounds. Hence she gets he is making a girl suit.

The moth is a direct clue and it leads her to something that reinforces the idea that Jame Gumb and John Grant are the same. If you recall when the moth fluttered and caught her attention, it acted like it was trying to land.

The thing is was trying to land on was spools of thread. Not the wooden kind a homemaker buys at a sewing shop, but the industrial type of spool set on a commercial sewing machine.

Surely the moth was the clue most of the audience got. And in an answer to this question one person suggested this ave it away. Yet if you recall the people Starling went to for validation of what type of moth it was had to cut it open to figure out it was the death's head moth. So she may have assumed, but that was still a stretch to a degree.

But the spools were the underline for me that this was THE clue.

Clarice does not go to the house knowing that the resident is Jame Gumb (remember Crawford told her the name of the individual who bought the equipment for housing moths). She goes to the house to fulfil Crawford's door-to-door instructions with people that knew Bimmel from her tailoring talents (Clarice found a dress with tape marking out the same diamond-shaped areas of skin which Bill takes from his victims; therefore, the clue is that he was in the same tailoring course perhaps?) At the very least, she goes looking for Mrs Lipman (not Littman as you all keep saying- it is 'lip'because of the film's centrality around orality, cannibalism, speaking/tongues etc) who was the teacher of sewing etc. She knows that its him when the rare moth flies into the disgusting kitchen and settles on the coils of cotton he has on the table. SIMPLE AS THAT

I think when Starling arrived at Lippmans house and Gum answered and invited her inside she began to get creeped out. Foster's demeanor immediately changed when she went inside "like a 6th sense". Also Gum asked the question if the "they" were close to catching someone, then mocked the police. Guilty people often ask about the investigation in a causal manner or even show up to the crime scene. and off course the plastic wrap and moth gave it away.

side note- just watch dateline or 48 hours and see how many guilty suspects voluntarily go to the police station and sit in an interview room for hours at a time just so they can find out what the Police know.