Considering Wilde's own predilection for aesthetics, and his tendency to be arrogant, we could sustain that Wilde shows pride more prominently than other deadly sins in the character of Dorian because pride is the one sin that Dorian can boast about without consequences.
Moreover, too much pride leads to arrogance, which also leads to overconfidence. This assurance that everything will go his way is what, ultimately, leads Dorian to his demise. Hence, using pride as a conduit of fate, karma, and consequences gives the author a great way to infuse a "moral" into a book, especially one which focuses entirely on sin. In fewer words, Wilde may have chosen for pride as its focus because a character like Dorian would undoubtedly have an extreme amount of it. Hence, that would be his weakest link.
Dorian is a man born with unique talents and gifts; he has personal beauty, charisma, money, a good position is society, and comes from a good family. Moreover, Dorian knows that he can get away with anything now: somehow, the key to eternal youth has manifested in Basil Hallward's picture of Dorian: the picture, and not Dorian, is aging and showing every sign of Dorian's wretched life.
Hence his pride is fed enormously each time that his beauty and youth weakens society into treating him like a form of demigod. Like a demigod, magical but imperfect, he wraps people around his finger, and lures them to his company with his amazing charm. However, we know that "something" always occurs after they have been lured in: Dorian has the capacity to shame everyone that he touches, and to bring them all to ruin.
I hear all these hideous things that people are whispering about you, I don't know what to say. Why is it, Dorian, that a man like the Duke of Berwick leaves the room of a club when you enter it? Why is it that so many gentlemen in London will neither go to your house or invite you to theirs? [..]Lord Staveley [...] said that you [...] were a man whom no pure-minded girl should be allowed to know[...]There was that wretched boy in the Guards who committed suicide. [...]. There was Sir Henry Ashton, who had to leave England with a tarnished name. [...] What about Adrian Singleton and his dreadful end? What about Lord Kent's only son, and his career?
When we see this pattern we cannot help but notice that it is pride and arrogance what leads Dorian to boost his own ego by destroying the dignity of others. He is like a black hole that, using his beauty and youth, disguises as someone good, and ends up swallowing them whole, until they become nobodies. In the end, Dorian will pay every consequence for his actions when he destroys his picture and the aging process reverts back to Dorian. It is then we we see that his looks and his youth, his money, and his position mean nothing in the end. He dies just like any other mortal does.