So there have been quite a few speculations about who inspired the characters in Yuri on ice.
As a figure skating fanatic, 1st I can say that the show is actually very well researched with a lot of cool touches for the figure skating fandom.
For example, Yuuri was training in Detroit before going back home, and Detroit (and Ann Harbor) is actually a big training center with a lot of international skaters training there year round or going there to get choreography. So it's the kind of detail that make perfect sense.
All the jackets are the exact replicas of the official jackets from the skaters from each countries, and the countries of the skaters showed so far make perfect sense. For example, when they show Viktor winning worlds, at the press conference the bronze medalist is from Kazakhstan which makes sense since a prominent skater from Kazakhstan -Denis Ten- has won world and olympic medals in the near past.
You can also see that in that picture :
Viktor has a poodle tissue-box behind him. That is a reference to Yuzuru Hanyu's very famous Winnie the pooh tissue-box (that actually set a trend among younger skaters to have their own customized tissue box).
Vicchan the poodle also looks a lot like Mao Asada's dog Aero who was publicly given to her by a sponsor and is kinda famous in the figure skating community since Asada is absolutely adored by a lot of figure skating fans, especially in Japan.
Besides, the person doing the commentator's voice in the 1st episode is Takeshi Honda a former skater who actually does/did some commentating for Japanese TV.
The choreographies in the show are also designed/supervised by Kenji Miyamoto who is a very successful figure skating choreographer who for example choreographed for former world champion Daisuke Takahashi or for Yuzuru Hanyu.
Here he is with the artist doing the drawings I think:
I am saying all that to show that the show is really well researched and that the figure skating fandom is actually quite happy in general with the show.
Even skaters have indicated watching the show. Here is a tweet of reigning world champion Evgenia Medvedeva about it (lol):
Now when it comes to the characters, if you follow figure skating a little bit, it is quite flagrant that the characters were all quite heavily inspired by real life skaters.
There have been some discussions about it, for example a lot of people feel like this:
And I partially agree but I don't think it's 100% accurate.
Viktor is the easiest, it's Evgeni Plushenko for sure. He is basically a figure skating legend. He is a 3 time world champion, participated in 3 olympics where he won silver, gold and then gold in the team event. He is still quite popular in Japan where he does a lot of ice shows and Yuzuru Hanyu, the reining olympic champion and Japanese superstar skater has been very vocal since he was a baby about his admiration for Plushenko and how he inspired him.
Here is Plushenko doing kabedon to Kenji Miyamoto (and comparison with Viktor):
Here is his merch vs Viktor merch in YOI:
Russian Yuri is quite easy too. He is a mix of Yulia Lipnitskaya and Artur Gachinski. One of the author actually stated that Yuri was inspired by Yulia and it is quite flagrant. She has the same stare/glare and look at those comparisons:
Yulia and Plushenko when they won team gold at Sochi olympics:
But Yuri also looks like Artur Gachinski and his skating career ressembles the one of Artur A LOT. Arthur was training alongside Plushenko, he was a very promising junior that had impressive debut as a senior skater. He was kind of the next Russian prodigy lined up to take over after Plushenko retired and he was also for some reason viewed as kind of sulky brat. (unfortunately he struggled with injury/motivation/personal problems after that and retired last year not being able to match the expectations for him).
Pics of Artur, including one where he was second behind Plushenko at Euros in 2012.
Japanese Yuuri is more difficult to ID. A lot of people say he is Yuzuru Hanyu but imo he only looks like Yuzuru physically (a bit). Like Yuuri, Yuzuru also worships Plushenko (Viktor). When he was little, he had a mushroom haircut trying to imitate Plushenko's mullet and he did and still does Biellman spin in his programs because Plushenko did.
Here is Plush giving flowers to Yuzu and Yuzu dying inside:
However, Yuzuru was always viewed as a prodigy. He won World bronze at 16 and olympic gold at 19, he now detains all world record scores for men figure skating. He never struggled with confidence and is a very aggressive competitor. So Yuuri's character doesn't ressemble Yuzuru at all.
However, another Japanese skater ressembles Yuuri
a lot: Tatsuki Matchida.
He was a late bloomer and he was 23 when he made it to his first Grand Prix Final. That final actually took place in Sochi just like in ep1, Machida finished last and then messed up at Nationals and didn't make the world team. He was training in America at the time and going to college, he struggled with self-esteem at this point too. (he then became a diva, made the olympic team, won silver at worlds and quit skating in shocking fashion in the middle of a season to go do a masters degree while his fans had a meltdown).
Here is Machida in action:
So there you go, my opinion:
Viktor is Plushenko
Yuri is Yulia and Artur mix
Yuuri is Tatsuki Machida with just a bit of Yuzuru.
if you read all of this, here is Yuzu and Artur for your effort:
) And it has the most life destroying ships to ever come out of an anime imo
quite unexpected while where on the topic whats your favourite junji ito short/series ? 
It's a favourite of mine!
ohh god please dont make me remember. 