This story starts with the Doctor attempting to drop Martha back at home after her extended "one trip" with him, but a news flash about an experiment for which Martha's sister is handling the PR intrigues him, so although he pops off in the TARDIS, leaving Martha virtually in tears, he pops right back again saying "Sorry, did he say he was going to change what it means to be human?" Given the Doctor's love of humanity (why else does he hang around Earth so much with the whole of time and space as his playground), that was bound to catch his attention ! It seems that Professor Richard Lazarus (Mark Gatiss) has discovered a way to rejuvenate human beings, thereby making them virtually immortal but, as is usually the case with immortality, it comes at a terrible price. Meanwhile, a mysterious figure, representing the same Mr. Saxon who's funding the experiment, tells Martha's mother that her daughter's new best friend is very dangerous. For some reason that I couldn't fathom (even after watching the episode three times!), Mrs Jones believes the man who tells her this and gives the Doctor a slap; at least the last time he got slapped by a Companion's mother (Jackie Tyler in Season 1's "Aliens of London), she had the excuse of the Ninth Doctor having kept Rose away for 12 months rather than the 12 hours he'd thought they'd been gone. The Tenth Doctor actually brings Martha back after a mere 12 hours away (that encompassed 3 adventures: "The Shakespeare Code", "Gridlock" and Daleks in New York), but gets a slap anyway - which seems rather unfair. I can't understand Mrs Jones' hostility at all. It's not as if the Doctor looks dodgy - far from it, actually, since he's wearing his tux !
Anyway, Lazarus' experiment goes badly wrong, leaving his body to undergo a genetic mutation that turns him into a huge, ugly scorpion-esque creature that proceeds to drain people of life and rampage around the Lazarus Labs building. The Doctor thinks he's killed it by "reversing the polarity" (a nod to Jon Pertwee's Third Doctor) of the device that Lazarus had used to rejuvenate himself, whilst he and Martha are hiding in the device, the Lazarus-monster is outside but turns the machine on with them inside it. Alas, Lazarus isn't dead and he drains the two paramedics who were trying to take his body away, and then holes up in Southwark Cathedral, a place he knows well as he used to shelter there as a boy during the Blitz. He and the Doctor have a philosophical discussion about longevity (which they'd already discussed after Lazarus rejuvenated himself). The Doctor says that facing death is part of being human, but Lazarus contradicts him saying that "avoiding death, that's being human. It's our strongest impulse." So the Doctor tells Lazarus
A long life isn't always a better one. In the end you just get tired. Tired of the struggle. Tired of losing everyone that matters to you. Tired of watching everything turn to dust. If you live long enough, Lazarus, the only certainty is that you'll end up alone.
Some fans see this as yet another reference to Rose, but I don't think it's just that. It's a reference to the fact that the Doctor is the last of the Time Lords, he's lost his whole family and his entire race, in the last few years, not just Rose. And Gallifrey is gone - as he so eloquently told Martha at the end of "Gridlock". And I think the Tenth Doctor's very tired of the struggle - especially against the Daleks (in the Dalek 2-parter he comments of the Daleks "They always survive and I lose everything."), but also the struggle against everything else that keeps trying to destroy the universe/humanity.
I liked Martha in this episode - from her glee at over-riding the Lazarus Labs' security so that she get everyone out, the fact that she insists on going back for the Doctor when he's the last person left in the building (which is very reminiscent of Rose insisting on going back to Satellite Five for the Ninth Doctor in season 1 finale "The Parting of the Ways"), to her insistence on going with the Doctor after Lazarus escapes from the ambulance into the Cathedral (and this in spite of her mother's objections), and the fact that she actually refuses to go with the Doctor again for just one more trip, pointing out that he isn't being fair to her and accusing him of seeing her as just a passenger (which he denies she ever was). The way the Doctor gives in so easily to her objection proves that he still wants her along, but he had to give her the chance to go back home and see her family, and the chance to decide not to go on travelling with him.