There are 3 spots for NA, they goto the winner of the summer split, the team with the most combined points from spring and summer split and the team that wins the gauntlet. C9s chance is in the gauntlet.
Since C9 did so good in spring and netted 70 points, even with 0 points from the summer they lock in 5th place in points. The way the gauntlet works, is the #5 & #6 team play, winner plays the #4 team, winner of that plays the #3 team and winner of that goes to worlds as the gauntlet winner. Gravity has locked in #6 and C9 have locked in #5, which means they will have to win 3 consecutive series (gravity, #4 team, #3 team). If liquid beats impulse, #4 team will be impulse, #3 team will be CLG, if impulse beats liquid #4 team will be Liquid and #3 team will be impulse. (I'm leaving out the scenarios where CLG beats TSM, because we don't live in a world where CLG beats TSM. If they did though Liquid/Impulse would be #3/#4 which one in which spot depends on who wins the 3rd place series but as far as C9 is concerned it doesn't really matter they would have to beat them both). Basically #4/#3 is going to be liquid/impulse unless both TSM & Liquid win, then (in combined points from the 2 splits) liquid would take 2nd and CLG would be 3rd and it'd be Impulse/CLG #4/#3
In summation, C9 will have to win the gauntlet, beating Gravity, Impulse, and liquid, with fairly high chance (TSM will beat CLG, and I think Liquid looks better than Impulse), CLG could be swapped for Liquid in the #3 spot.
They actually have a fairly good chance I think, in NA the teams are really hit or miss during the playoffs. The only exception to that really is TSM who are hit or miss during the season, but always show up for playoffs, and they've already secured #1 if they win or #2 if they lose, so there is no scenario where C9 has to beat TSM in the gauntlet. Remember all these teams in the gauntlet are coming off losses in playoffs to, except C9 who didn't make playoffs and are only going due to their spring performance... that means C9 should have a fairly clear roadmap of how to beat each of these teams, on top of which, there is a good chance they will be tilting after losing playoffs.