These murderous adventures were not forays undertaken to avenge injuries, nor to settle old disputes or sudden fallings out; no, as a rule they were simply duels between strangers. (3.1)
The passage strongly condemns the Knights of the Round Table and their epic fail as far as real justice goes. They're not really knights in that they don't protect the people or do good deeds—dudes just fight each other.
The nation as a body was in the world for one object, and one only: to grovel before king and Church and noble. (8.4)
Don't be shy, Hank the Yank: tell us what you really think.
My works showed what a despot could do with the resources of a kingdom at his command. (10.3)
The Yankee sports a real ego sometimes, but he also stresses the good he can do against unjust institutions like the Church once he's in charge. That's a pretty important part of his character—if he didn't believe that he could change a whole country from the ground up, he would never go forward with any of his plans.
She slipped a dirk into him in as matter-of-course a way as another person would have harpooned a rat! (16.6)
It's not just that Morgan Le Fay kills the page (for bumping into her no less)—it's that she doesn't give a second thought about doing so. (It's apparently okay—or at least more okay—to kill someone and feel bad about it afterwards.) Morgan's a villain, so we kind of expect evil behavior from her, but it still happens an awful lot in this book.
I will say this much for the nobility: that, tyrannical, murderous, rapacious, and morally rotten as they were, they were deeply and enthusiastically religious. (17.1)
A little irony: it's good for your blood. Hank points out that religious enthusiasm seems to balance out all sorts of crimes… which is another way of saying that religion is crime's enabler. C'mon dude, it'll be totally cool to beat that turnip farmer to death. We'll be absolved of our sins at church…
A gentleman could kill a free commoner, and pay for him—cash or garden-truck. (18.4)
Peasants are equated with property, meaning it's okay to murder them if you can pay the fine. Hank ironically points out a couple of ways to do it… and his fake casualness about it makes it seem all the more horrible.
The deer was ravaging the man's fields, and he had killed it in sudden passion, and not for gain. (18.2)
The quote refers to a man in Morgan Le Fay's dungeons, sentenced to die for killing a deer. Hank uses a ridiculous legal argument to free him; is such absurdity justified, or does it simply demonstrate how hard it is to find true justice?
The king's judgments wrought frequent injustices, but it was merely the fault of his training, his natural and unalterable sympathies. (25.2)
Hank gets to the root of injustice here, marking it as a societal problem, rather than just a kill-the-bad-guy problem. You don't see that very often in stories, and instead they usually focus on getting the villain as a sort of one-stop cure-all. Twain knew better though, and understood that real change takes time and effort.
"If I do a thing which ought to deliver me to the stocks, and you know I did it and yet keep still and don't report me, you will get the stocks if anybody informs on you."
"Ah, but that would serve you but right," said Dowley, "for you must inform. So saith the law." (33.13-14)
Why don't the peasants just gang up on the nobles and demand justice? This exchange gives us the answers.
Slavery was dead and gone; all men were equal before the law; taxation had been equalized. (40.1)
This pretty much marks the start of the big downward slope that turns the book from a comedy to a tragedy. Hank gets so close to achieving real justice before the ending washes it all away, and Twain wants us to know how close that "close" really is. It makes the ending sting that much more.