As the are examples in other answers regarding fear and pity in the play, I will focus my answer on the issue of hatred in Macbeth.
Other than Macduff's hatred for Macbeth for killing his family, there's remarkably little outright expression of hatred in the play. The word "hate" appears only twice in the play, spoken once by Banquo in passing reference to incurring Macbeth's displeasure [1.3.64] and once in reference to feelings towards Macbeth by the people as a whole [5.2.15]. The word "hateful" appears only once, spoken to Macbeth by Young Siward [1.7.10–11], who Macbeth promptly kills in battle near the end of the play.
Otherwise, the murders that Macbeth commits, and those that he orders, are not committed out of hate, but as part of Macbeth's strategy to eliminate potential rivals. In the case of the murders of Lady Macduff and her son, this was Macbeth's way of sending a message to Macduff that he is not to threaten Macbeth's place on the throne.
A notable exception to this lack of the outright expression of hatred in the play is Lady Macbeth's unexplained, obsessive hatred of Duncan.
At the beginning of act 1, scene 5, Lady Macbeth is reading the letter that she just received from Macbeth telling her about the Witches's prophecies, one of which—that Macbeth would be Thane of Cawdor—has already come true. The other prophecy, in Macbeth's words, "Hail, King that shalt be," has yet to be fulfilled. This starts Lady Macbeth thinking about what she can do to help fulfill the Witches's prophecy.
Then, a Messenger arrives:
LADY MACBETH: What news do you have?
What is your tidings?
MESSENGER: The King comes here tonight.
LADY MACBETH: Thou'rt mad to say it! [1.5.26–29]
Lady Macbeth can't believe it. The King is coming to her own castle! It's the perfect opportunity for Macbeth to fulfill the Witches's prophecy, by killing Duncan:
LADY MACBETH: The raven himself is hoarse
That croaks the fatal entrance of Duncan
Under my battlements. [1.5.39–41]
What provoked such a fierce reaction to Duncan's visit? A few minutes ago, all Lady Macbeth knew was the Witches's prophecies, and now she's ready for Macbeth to kill Duncan and become King.
Then Lady Macbeth launches into one of the most famous speeches in all of Shakespeare's plays:
LADY MACBETH: Come, you spirits
That tend on mortal thoughts, unsex me here,
And fill me from the crown to the toe top-full
Of direst cruelty! make thick my blood;
Stop up the access and passage to remorse,
That no compunctious visitings of nature
Shake my fell purpose, nor keep peace between
The effect and it! Come to my woman's breasts,
And take my milk for gall, you murdering ministers,
Wherever in your sightless substances
You wait on nature's mischief! Come, thick night,
And pall thee in the dunnest smoke of hell,
That my keen knife see not the wound it makes,
Nor heaven peep through the blanket of the dark,
cry 'Hold, hold!' [1.3.41–55]
Later, Macbeth tells her the same thing about Duncan's visit:
MACBETH: My dearest love,
Duncan comes here to-night.
LADY MACBETH: And when goes hence?
MACBETH: Tomorrow, as he purposes.
LADY MACBETH: O, never
Shall sun that morrow see!
Lady Macbeth and Duncan have a backstory that might explain Lady Macbeth's over-the-top reactions to Duncan's visit. The backstory doesn't appear anywhere in the play itself, but it would very likely have been known to Shakespeare from the sources he used in writing Macbeth.
There was a real Macbeth, a real King Duncan, and a real Lady Macbeth, whose name was Gruoch. Long story short, Gruoch had a hereditary claim to the throne of Scotland through her grandfather, Kenneth III. Gruoch fell in love with Macbeth, but when Duncan became King, he ordered Gruoch to marry Duncan's cousin Gillecoemgain—strike one.
As a side note, Gruoch and Gillecoemgain had a son name Lulach. Lady Macbeth makes a single cryptic reference to having a child when she's trying to bully Macbeth into killing Duncan in act 1, scene 7:
LADY MACBETH: I have given suck, and know
How tender ’tis to love the babe that milks me [1.7.60–61]
Now let's go back to the backstory. Duncan, who was not quite the saintly old king of Shakespeare's play, ordered Gillecoemgain to be killed or possibly killed Gillecoemgain himself—strike two.
Gruoch was forced to flee from Duncan with her son (strike three), and she fled to Moray, a part of Scotland ruled by Macbeth (not far from Macbeth's castle at Inverness in the play), and she finally married Macbeth, the love of her life. Macbeth later killed Duncan in battle (not by murdering him in his sleep as in Macbeth), making Macbeth king, and Gruoch queen.
This backstory could be the reason that Lady Macbeth hated Duncan so much and why she was so determined to get revenge on Duncan and have the Witches's prophecy fulfilled.
And all this time we thought it was just her raging ambition. . . .