And flights of angels sing thee to thy rest.”
 
In the wake of Hamlet's death, Horatio's kind words for Hamlet show just how much he cares, how much he has always cared. These words are also the basis of most Hamratio pairing.

Excerpt From: Shakespeare, William. “Hamlet.” Simon & Schuster. iBooks. (5.2.397-98)

Excerpt From: Shakespeare, William. “Hamlet.” Simon & Schuster. iBooks. 5.2.413-28)

 
Here, Hamlet is dying, and Horatio is ready to kill himself because of this. Only by Hamlet's command does he not. This shows more about how much he cares about Hamlet, with a little Shakespearean flair.

Excerpt From: Shakespeare, William. “Hamlet.” Simon & Schuster. iBooks. 5.2.373-74)

Excerpt From: Shakespeare, William. “Hamlet.” Simon & Schuster. iBooks. 5.2.434-40

Here Horatio reveals his skepticism about the ghost, but also his skepticism in general, which manifests throughout the play.

Excerpt From: Shakespeare, William. “Hamlet.” Simon & Schuster. iBooks. 1.1.35

Here Horatio describes King Fortinbras. Unlike Hamlet, Fortinbras has mettle and stomach. This is very important. See, Hamlet is not just about the theatrical bloodbath and revenge, but in fact more centered on Hamlet's hesitation to get revenge as well as the question of whether or not revenge is worth it. So Horatio's juxtaposition between Hamlet and the noble Fortinbras sets the stage for a not-so-vengeful approach.
  

Excerpt From: Shakespeare, William. “Hamlet.” Simon & Schuster. iBooks. 

1.1.107-16

Or to the dreadful summit of the cliff

That beetles o'er his base into the sea, And there assume some other horrible form,

Which might deprive your sovereignty of reason

And draw you into madness? Think of it.

The very place puts toys of desperation,

Without more motive, into every brain

That looks so many fathoms to the sea

And hears it roar beneath."
1.4.72-81

While Hamlet is reckless, Horatio holds him back with reason and concern. Horatio worries that the ghost will use its unearthly powers to cause Hamlet to go mad. Considering that after this he indeed goes mad, it can be argued that Horatio was right all along.

Horatio, you’re the best man I’ve ever known.
HORATIO Oh, sir—
HAMLET

Don’t think I’m flattering you. What could I hope to get from you, who’ve got nothing but your charm to support you in life? Why would anyone flatter a poor person? No, keep flattery for kissing the hands of those who can pay well. You understand? Ever since I’ve been a free agent in my choice of friends, I’ve chosen you because you take everything life hands you with calm acceptance, grateful for both good and bad. Blessed are those who mix emotion with reason in just the right proportion, making them strong enough to resist the whims of Lady Luck. Show me the person who’s master of his emotions, and I’ll put him close to my heart—in my heart of hearts—as I do you. But I’m talking too much." 3.2.45-58

This is quote shows how Hamlet truly thinks the world of Horatio. Because Horatio is, in Hamlet's eye, poor, Hamlet has political reason to hang around Horatio, making their friendship more genuine. This is a clear opposite to Hamlet's fake friendship with the esteemed Rosencrantz and Guildenstern, who betray Hamlet. (Fake friendship versus true friendship, by the way, is an important theme in most of Hamlet's works). This is also important because what Hamlet values in Horatio --emotion checked by logic -- is also what Hamlet hates the most in himself.