It doesn't open very seriously, does it? It's slightly surprising to see Romana regenerated for no obvious in-story reason, and the manner in which the regeneration is treated is at odds with how it's been treated before. In the past, regeneration has been treated as a last resort, with one body essentially dying and being replaced by another, but here it's a much more casual affair, with various bodies being presented in the same manner as which Tom Baker presented various costumes in Robot.
Yet after this opening scene (which also features K-9 with laryngitis, somehow108), Destiny of the Daleks is a surprisingly serious affair. The scenes of the Doctor and Romana clambering over the ruins of Skaro (not that they know it's Skaro yet) are very atmospheric, and there's a sense of dread that pervades everything. Well, almost everything; the bit with the Doctor trapped under a pillar and reading a book about the origins of the universe (complete with a Hitch-hiker's Guide to the Galaxy in-joke from new script editor Douglas Adams) is pretty jokey. And since this is a Terry Nation Dalek story, we have to wait until the cliffhanger for the Daleks to make their appearance, as they threaten Romana.
The second episode is just as serious; the Doctor appears to be taking the threat of the Daleks very seriously, and he keeps making noises about how he thinks he knows what it is the Daleks are searching for in the ruins of the Kaled city, but he won't actually tell us what that is. "I'll tell you when I find out," he says. And while he's investigating the ruins, trying to find whatever the Daleks are looking for before they do, Romana is subjected to an interrogation by the Daleks (and looks suitably terrified by them -- you can even see a tear on her face when they finally stop questioning her) and then forced to help them excavate the ruins. It's honestly pretty brutal, and the atmospheric direction from Ken Grieve (helped by this new technology they're trying out called "Steadicam") helps a lot with this. And while there's another jokey moment in episode two (the famous bit with the Doctor taunting a Dalek: "If you're supposed to be the superior race of the universe, why don't you try climbing after us?"), for the most part this somber air pervades the whole thing. Oh, and look: we get our first localized extermination effect (as opposed to the whole picture going negative) -- it's quite a nice effect.
And it turns out that the Daleks are looking for Davros (even though it looked like they killed him at the end of Genesis of the Daleks -- and was the Kaled bunker really under the city in that story?) -- who, the cliffhanger shows, apparently isn't as dead as you might think...
And incidentally, full marks to whoever pulled out the background sound effects from the first Dalek story for use in this story.
108 The real world reason being that John Leeson has left as K-9 and has been replaced by David Brierley -- but still, they couldn't come up with a better reason than laryngitis?