Exposition (Initial Situation)
We see into the mind of the spy who reports the Union position to Lee and Longstreet. Harrison (that's the spy) reports to Lee, while on the Union side, General John Buford's cavalry help hold back the Confederates so that the Union Army can arrive on time.
Rising Action (Conflict, Complication)
Thanks to Buford's defense, the rebels aren't quite in the position they'd like to be in. Lee wants to capture high ground in Gettysburg, but the generals he orders to do this, Richard Ewell and Jubal Early, fail to act. The Union ends up controlling two crucial hills—Cemetery Hill and Culp's Hill—and enjoying the view.
That's how the situation stands at the end of the first day, but there's still more high ground to take. On the second day, the Union troops, including Joshua Chamberlain's, take even more of it, and they end up controlling a rocky hill called Little Round Top. Lee orders Longstreet's soldiers to try to flank the Union army by taking Little Round Top.
Climax (Crisis, Turning Point)
In some ways, The Killer Angels has two climaxes—one for the Union and one for the Confederacy—but since Chamberlain is the moral center of the book, you should probably pick the Union's climax if you need to pick one.
For the North, the climax comes with Chamberlain's tenacious defense on Little Round Top during the second day of the battle. During this battle, Longstreet's troops fail to gain the high ground. When Chamberlain's men start to run out of bullets, he orders them to charge with their bayonets—converting defense into offense. This prevents the Confederates from taking Little Round Top, and, it basically wins the battle for the North and paves the way to ultimate victory in the war itself.
Chamberlain pretty much saves the day—all that's left for Lee and Longstreet is the Confederate climax: Pickett's famously doomed charge on the third and final day of the battle.
After Pickett's division is virtually obliterated, and after General Armistead dies while sadly betraying his vow to his old friend Hancock, all that's left is for Lee and Longstreet to pick up the pieces. Lee rides through the tattered remnants of Pickett's troops, admitting out loud that it was his fault, and he's to blame for the disaster. Meanwhile, Chamberlain survives the ferocious rebel artillery bombardment and watches the destruction of Pickett's charge.
After the Confederacy loses the battle, Longstreet doesn't know if he can still lead men into battle for a doomed cause. Lee admits that their cause has been destroyed: they have nothing to fight for now but the army itself, and that's no way to win. While watching the Confederate assault, Chamberlain is impressed by their courage and the tragedy of their failed attempt. He marvels at the carnage. But afterwards, as he and his brother Tom talk the battle over, they agree that the war ultimately isn't about states' rights or the other things the South claims to be fighting for: it's about slavery.