How does Monte Cristo avenge the treachery of his friends?The Count of Monte Cristo by Alexandre Dumas

In the chapter (49) entitled "Ideology," of The
Count of Monte Cristo,  Monte Cristo has a rather intimate encounter with de
Villefort in which he accuses de Villefort,  "Do you really think that what you do
deserves being called anything?"  When de Villefort is, of course, insulted, their
conversation turns upon Monte Cristo's raison d'etre.  In response
to the Procureur du roi's remark that he unquestionably has some ambition, Monte Cristo
reveals,

I,
too, as happens to every man once in his life, have been taken by Satan into the highest
mountain in the earth, and when there he showed me all the kingdoms of the earth, and as
he said befre, so said he to me, "Child of earth, what wouldst thou have to make thee
adore me?'  I reflected long, for a gnawing ambition had long preyed upon me, and then I
replied,  "Listen,--I have always heard tell of Providence, and yet I have never seen
him, nor anything that resembles him....I wish to be Providence
myself, for I feel that the most beautiful, noblest, most sublime thing
in the world, is to recompense and
punish.'

This passage is key
to Monte Cristo's methods of revenge against his enemies and reward to those he loves. 
For, he is neither simply the diabolical villain nor hero. Rather, he is more the
Providence of the Old Testament who wreaks "an eye for an eye" and rewards the just. 
With regard to his enemies, Monte Cristo uses what is their sin against them. For
instance, Danglars the banker is destroyed by his own greed when Monte Cristo plots a
series of financial disasters for him; de Villefort, the defender of French law, is
destroyed when his attempted burial of his illegimate child is exposed in court; the
Count de Morcef (Ferdinand Mondego), the "peacock" who glories in his rank and fame, is
exposed as a traitor who has sold an ally to the enemy. Caderousse, the thief, is slain
by his fellow thief, Benedetto (Andrea de Calvacanti) whom he
betrayed.

While Monte Cristo arranges for his enemies'
destruction, he does not kill them.  Instead, he simply arranges for their own sins to
destroy them, as Providence would.  Likewise, he intervenes as Providence would for the
good and worthy by providing Monsieur Morrel the financial aid that he needs in order to
stave off ruin.  Monte Cristo intervenes to save Valentine de Villefort and enable her
to marry the son of his beloved friend, Maxillian
Morrel.

While Monte Cristo's sin is one of pride in feeling
that he is an agent of Providence, he achieves his redemption in the end of the novel. 
In a letter to Maxilmilian:

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Tell the angel who is going to watch over you,
Morrel, to pray for a man who, like Satan, believed for the moment he was the equal of
God, who now acknowledges in all Christian humility that in God alone is supreme power
and infinite wisdom....which is contained in these words:  Wait and
hope!