What is the ultimate goal of the murky goings on in Sicario?

As explained by Matt (Josh Brolin) to Kate (Emily Blunt), the purpose of the task force is to "take down" Mexican drug lord Fausto Alarcón. While this can be construed as "arrest and convict", the actual plan is to kill him, plain and simple.

The CIA's plan is that by weakening the Mexican cartels by such killings, the Colombian cartels can once again assume control of the drug trade in the americas. Their argument is that back when the Colombian cartels had a near-monopoly on the trade, there was a "semblance of order".

This period of time is referred to as "Medellín" in the movie, after the Colombian city and cartel of the same name. (Of course, the irony is that the United States did a lot to bring an end to the Medellín cartel in the first place.)

Since those "good old days", however, Mexican cartels have taken over, and the violence and chaos of the drug trade has both grown and moved much closer to the United States - and spilled over.

So, in essence, it's the CIA choosing to side with the devil they know. Like supporting a dictator or even overthrowing an existing regime, in order to provide "stability" from a U.S. point of view. Or, in this case, at least move the problem farther away.

There's also the characters' personal reasons for wanting Alarcón dead. Alejandro (Benicio del Toro) was a public prosecutor in Juárez, Mexico. His wife and daughter were killed (horrifically) by Alarcón's group. Since then, Alejandro has become a mercenary - or vigilante - in his attempt to get revenge.

As Matt explains, Alejandro will work for "anyone who will let him of the leash". That is, help him to kill the people responsible for the killing of his family. In this case, the ones letting him off the leash are the United States and, indirectly, the Colombian cartels. Hence why another character refers to Alejandro as "Medellín": He's doing their dirty work.

Macer's (Blunt) goal was to get back at those ultimately responsible for the bodies in the house raid at the beginning of the film.

She's really only there so that the CIA can operate on US soil, for which they need an FBI agent along for the ride.

Gillick's (del Toro) goal was revenge for the killing of his wife and daughter by Alarcón.

Graver's (Brolin) goal was to help Gillick to kill Alarcón. He did this in order to help consolidate power in the hands of Colombian drug lords at the expense of the Mexicans. He did that to help move the drug war away from the US/Mexican border.

As far as I understand, the assassin who ended up killing the drug lord was the old leader of a drug cartel in Colombia. When Mexicans get in American drug market and get the all market share, he became useless and he made a agreement with US against Mexican cartel. On the other hand, the drug lord killed his child and wife. So he wanted to take his revenge and US wanted to dismiss the Mexican cartel. And this was the most easy way to kill the leader because they know in these kinds of group when the leader disappears, the hierarchy is disrupted and they start a war between themselves.