In "My Last Duchess" it is obvious the Duke had the Duchess murdered. Is the Duke trying to justify the murder or was he boasting about it?

You're right; it is pretty clear the Duke murdered his former wife, the Duchess.  He might be trying to justify it and he's certainly not ashamed to admit the deed, so he's possibly boasting about it.  I'd like to present a third option--he's warning his NEXT Duchess about what is expected of her.  Both the audience to whom the Duke is speaking and the sights they visit are significant in this poem. Here's the evidence:

The Duke is escorting an emissary from his future wife's father (he says to him, "the Count, your master") and giving him the grand tour, apparently to make the final marriage arrangements on behalf of the Count.  Funny how he stops only two places. One is the portrait of the former Duchess and one is his impressive statue created by a famous artisan, of Neptune, a god, taming a sea horse. 

First the painting.  He says to the unnamed emissary, "That's my last Duchess."  Read it this way:  That's my LAST Duchess--you know, the one who didn't put me above others, who valued a cherry tree branch as much as me, who refused to show me proper deference and eventually the one whose "smiles stopped altogether."  It's a clear warning--my NEXT Duchess, he implies, might want to pay attention to the things which displease me, and I expect the message to be relayed by you, Mr. Ambassador, in the appropriate manner.

The final stop on their tour is a pause before the Innsbruck statue.  The picture here is the same--I am master/god of my domain and those around me must be tamed. 

He's neither embarrassed nor is he apologetic regarding the demise of the former innocent (and apparently lovely, both inside and out) Duchess.  Her only fault, at least from the Duke's description, is that she failed to show the proper deference to his name and position.  The NEXT Duchess will certainly need to remember this lesson if she doesn't want to be the his "Next Last Duchess."