Technically, Aeneas himself is responsible for Dido's suicide, as his abrupt departure from Carthage in Book IV drives the queen to despair. However, as the previous educator has noted, the true responsibility lies with the gods rather than Aeneas.

When the Trojans arrive in Carthage, Dido welcomes the refugees warmly and invites them to live in her city as long as they wish. Aeneas's mother, the goddess Venus, worries that her fellow goddess Juno will try to turn Dido's mood against the Trojans, and so she arranges for Dido to fall madly in love with Aeneas to ensure his safety. Speaking to her son, Cupid, Venus says:

I fear the outcome of Juno’s welcome here . . .
She won’t sit tight while Fate is turning on its hinge.
So I plan to forestall her with ruses of my own
and besiege the queen with flames [of love],
and no goddess will change her mood—she’s mine,
my ally-in-arms in my great love for Aeneas.

Dido is enchanted by Cupid and falls desperately in love with Aeneas. Juno, who hates the Trojans, is annoyed by Venus's ploy to protect them. She turns Dido's love to her advantage, however, by using it as a snare to trap Aeneas in Carthage, thus preventing him from achieving his destiny of founding Alba Longa. If Juno cannot hurt Aeneas directly, she reasons, she will interfere with his fate instead. She arranges for Dido and Aeneas to become stranded in a small cave during a storm, where they act on their mutual desire for each other and sleep together for the first time. This consummation of their lust is intended by Juno to be a "wedding" binding the two mortals together, and indeed Dido considers it her "marriage" to Aeneas:

From now on, Dido cares no more
for appearances, nor for her reputation, either.
She no longer thinks to keep the affair a secret,
no, she calls it a marriage,
using the word to cloak her sense of guilt.

Aeneas and Dido are happy together until word reaches Jove that Aeneas has seemingly abandoned his destined course for Italy in favor of staying in Carthage with his new lover. Jove is angered by this news and dispatches Mercury to go directly to Aeneas and remind him of his destiny.

Mercury lashes out at once: “You, so now you lay
foundation stones for the soaring walls of Carthage!
Building her gorgeous city, doting on your wife.
Blind to your own realm, oblivious to your fate!"

Aeneas is stunned by this criticism and appalled to realize he has been neglecting his duty to the gods by staying in Carthage. Although he loves Dido, it is his fate to sail to Italy, and now that Mercury has reminded him of this, he knows he must leave Carthage at once. He's sad to leave Dido, but Aeneas is a very pious man—his duty to the gods overrules any personal feelings he may have about that duty. He worries about how to tell Dido he is leaving, even as he makes preparations to go:

Yet he himself, since Dido who means the world to him
knows nothing, never dreaming such a powerful love
could be uprooted—he will try to approach her,
find the moment to break the news gently,
a way to soften the blow that he must leave.

Dido begs Aeneas to stay with her, for the sake of their love. Aeneas, in a rather callous bit of rules-lawyering, says that he never actually married Dido and therefore does not owe her the loyalty a husband owes to a wife. In any case, the gods have commanded him to leave, and so he must. Dido screams a curse down on his head for breaking her heart, but it's to no avail:

Aeneas
is driven by duty now. Strongly as he longs
to ease and allay her sorrow, speak to her,
turn away her anguish with reassurance, still,
moaning deeply, heart shattered by his great love,
in spite of all he obeys the gods’ commands
and back he goes to his ships.

Broken by her grief and despair, Dido orders a pyre built of all Aeneas's things. She climbs onto the pyre and kills herself with a sword in full view of Aeneas's ships. The pyre is then burnt, but Aeneas has been urged by Mercury to make sail immediately, and so he sails away unaware that his lover has killed herself for his sake. Dido blames Aeneas and holds him responsible for her love and her rage, but it is evident to the reader that the gods have interfered in these mortals' lives and led to Dido's tragic end.

On face value, Aeneas' departure is the reason why Dido goes mad and ends up killing herself.  It is his insensitivity and his failure to understand their connection and relationship after consummation that causes her to kill herself.  Yet, I think that the fault may lie in the Gods' actions.  Like Homer, Virgil is able to bring out how the temperament of the Greek immortals is actually more akin to mortals, who themselves must act like forces of divinity in impossible situations.  It is Venus who enables Dido to madly fall in love with Aeneas.  Additionally, it is the alliance between Venus and Juno that ensures Dido is going to be placed in a predicament where her sanity does not even register a concern.  It is Mercury who descends to warn Aeneas that his destiny is not to "build a woman's city" and encourages him to leave.  Of course, it is Jupiter who does not aid mortals in difficult predicament or at least provide guidance to them as to what to do in such agonizing of a condition.  In this, I think that a large level of Dido's death can be placed on the feet of the Gods', who use Aeneas as a way to carry out their own desires.