The first thing to say is that the two poems are both dramatic monologues, a style which is specifically associated with the Victorian era of English Literature - and more specifically, associated primarily with Robert Browning, who wrote both of the poems you mention. (The other big exponent of the dramatic monologue in this period is Tennyson, if you want to research this any further). So it might tell us something about the period that this particular style - a soliloquy, if you like, by a character who speaks directly to an auditor (and thereby to the reader) became popular.
Thematically, though there are a few other things to say. The first is that both of the speakers of these monologues are murdering psychopaths! The Duke, in My Last Duchess, has clearly finished off his last duchess in some sinister way he never explicitly states:
Oh sir, she smiled, no doubt,
Whene’er I passed her; but who passed without
Much the same smile? This grew; I gave commands;
Then all smiles stopped together.
Similarly, Porphyria's lover strangles Porphyria to ensure that - very similar to the Duke - he can possess her forever without her going away. And the way to do this is by - in both cases - killing her.
The Victorian era was fascinated by the macabre, and one of the things associated with it in popular memory is foggy, lamplit streets where murderers and weird deviants roamed at night. Just think of the number of sinister, crime-related stories to come out of this period: Sherlock Holmes, Dr. Jekyll and Mr. Hyde and so on and so on - not forgetting the real-world newspaper sensation of Jack the Ripper, or the emergence of crime magazines to a wider audience than ever before.
The very repressed Victorians seemed to love things related to deviant behaviour - whether sexual, criminal or just straightforwardly violent, and these two poems both allow the reader to travel inside the mind of a murderer.
The other thing it would be worth thinking about is the idea of atheism and questioning Christianity - neither speaker, of course, seems to worry about their behaviour in a religious light. Porphyria's lover even seems to mock God at the end of the poem:
And all night long we have not stirred,
And yet God has not said a word!
The Victorian period was a time of a major crisis in faith, often associated with the emergence of the science of evolution and Darwin's discoveries. It's interesting that religion in these poems does not feature, despite the fact that the poems themselves explicitly deal with issues of morality - and you might find that typical of this period of literature.
Hope it helps!
My initial reaction is to say, morality and sexual represession. In other words, these poems focus the reader on questions of morality and sexuality. In My Last Duchess, the speaker, a duke, murders his wife because he believes she was too friendly with people. There is a suggestion that she was friendly to the point of adultery, but again, it is just a mere suggestion. There is stronger evidence that in fact she was merely friendly and the duke is actually a controlling, manipulative, jealous psychopath. So the question is one of morality -- did the wife deserve to die? Is fornication (whether it actually occurred or not) a justifiable reason to kill someone? Porphyria's Lover takes up similar questions. It is suggested that Porphyria is not married to the speaker; she may just be his lover. Is her death acceptable because she was engaged in what may have been deviant sexual behavior (as she may have been married to someone else). Overall, I see these two poems as dealing with themes of sexuality and morality and the Victorians would have been very interested in these poems, precisely because people tend to be very interested in the subjects that society says is taboo.
Both of these poems reveal the murderous psyches of the speakers and their sick motivations for killing their female lovers -- which lead the reader to question the morality and justice of the murder.