Looking back over the core Kripke years, one realizes that it wasn't just about the brothers, but 5 years also battling their arch nemesis. An arch nemesis so fiendish, he continued to torment them for years after his death (not unlike the original Green Goblin and his son).
Of course, because of the factors that go into making a show, there will arise some incongruities in long running arcs. But that's no fun! Instead, let's imagine that everything really was airtight and recreate what Azazel's full plan was from start to finish.
So apparently from Day 1 Azazel has wanted to free Satan. To do so, he would need:
1) A righteous man to shed blood in hell.
2) Break 64 other seals.
3) Have a human would choose of their own free will to kill - with demonic power - the first born of Lucifer: Lilith.
3 would probably be the easiest to accomplish for Azazel. Just give a child a taste of demonic power and then a reason to kill Lilith. Letting them meet her for more than ten minutes should suffice. However, it will be more effective if the child is "infected" young and grows up with the power in his bones. But you know how spiritual bureaucracy is; he can't just walk into people's houses and feed blood to their kids. Thus he spends several years (particular in the 70s) making deals with people to allow him to return at their child's 6-month birthday.
While working on this, Mr Yellow-Eyed also gathered a host of demons to follow his command. An army of his own on earth would make step 2 considerably easier and it could distract the forces of heaven long enough for him to free the big bad himself.
Now let's fast forward to 2006. John Winchester has sold his soul. Some months later, Azazel begins gathering his "special kids" for a showdown to see who is the most powerful. From the clues we have John's probably been in hell for 30-50 years (hell time), meaning Azazel is confident that John will break at any moment, accomplishing #1 and setting everything in motion. So once he has determined which person is going to be accomplishing goal 3 (Jake in this case), Azazel has the guy open up the demon's gate to let out the army and get it started on goal #2. However... the Winchesters have to go and screw everything up, toasting the Yellow-Eyed Demon.
Now what about Lilith? If she agreed on the plan, was even willing to sacrifice herself to free Satan, why did she try to kill Sam so much in season 3? After much study the answer appears to be: because the initial plan was scrapped. At the end of season 2, John had escaped hell, meaning there's no righteous man down there to break the first seal. So why not settle for good 'ole world domination as a consolation? May as well get Sam & Dean out of the way too. Yet what do we find at the end of the year? Not only have the brothers' efforts to rescue Dean failed, but so did her effort to blow up Sam. This gives Lilith 2 things she needs: A righteous man in hell (again) that can break the seal and confirmation that Sam really does have the mojo to kill her eventually. As well as a hankering to do so. Thus we come to season 4. Lilith goes for the goal, and unknown to anyone, the angels only put up a token resistance because they're ready for the end-game as well.
Yet in the end, Azazel still won.
FAQ (that is, the most common questions I get related to this plan)
Why did the YED 'infect' so many kids in the first place?
Ever heard "don't put all your eggs in one basket"? In the season 2 episode Hunted, we even have a hunter named Gordon running around, killing the special kids. If the angels realized what the plan was, they might get their smite on as well. It doesn't take a lot of foresight (only common sense) to realize this was a likely possibility (much less once Dean revealed that the angels were keeping watch) and that you might want some extra kids on hand just to avoid setbacks.
So why have the competition? Why not use all the kids?
1st factor) Remember that Azazel isn't liked <em>by anyone</em>. 2nd factor) Recall that in season 4 we see that as Sam grows in power, he gains the ability to kill demons with his mind. Now we don't know Azazel's power compared to Alastair or Lilith. However if one human could eventually get the power to kill either of those two, then carting around a dozen or so of these brats (all of whom are longing to off the tyrant at the first opportunity) would probably end up with him nuked in short order. 1 person he can probably handle. More than that poses an unacceptable risk.
Still, why have a competition at all? Why not just pick a kid at random?
A common theme running throughout Supernatural is the intelligence of hunters (and humans in general). Azazel himself explains that he wanted Sam on the road, gaining the training and smarts needed to lead the army of the damned. Now keep in mind the worst case scenario of someone trying to stop his efforts (which did happen at the close of season 2) especially heavenly someones as Dean's time junket gave YED a clue that the angels would start fighting back at his plan. Therefore if you need something important done - something that a lot of people are going to be trying to prevent - you'd want the smartest, strongest, most capable person pulling it off. In this case, the competition was Azazel's method for determining which kid would be best able to handle letting the army out, then leading it to break the seals even against angel interference.
(these are the most common questions I've been asked offline, any others?)