Deep Breath is, at its core, a fairly typical regeneration debut story. The story itself wasn't bad, but wasn't exactly anything special either. But a regeneration story has the primary function of successfully introducing the new Doctor to the audience. This is the single most important thing we needed from Deep Breath, to break in Peter Capaldi as the new Doctor, and, as such, the episode did an excellent job of it. Everything else is secondary, in my opinion, although there were definitely both plus and minus points.
As I've said, the story was nothing exceptional, not exactly setting the world on fire for me. However, there were some nice scenes in it. The character set-pieces and interaction were generally excellent (with the exception of one thing). There were also quite a few nice bits of humour in among all the sad bits, and I also really liked seeing Lady Vastra, Jenny and Strax again. I enjoy these characters a lot, and I think that Strax is absolutely hilarious.
On the negative side, the plot itself was definitely a bit thin, there were at least a couple of glaring plot and character inconsistencies, particularly Clara's aggressively negative overreaction to the new, older Doctor, which was TOTALLY out of character. Any other companion, yes, it might've actually been more realistic, but not the Impossible Girl. She's met them all. And even if, as some people argue, she has no memories of her other lives (which I disagree with), her reaction was STILL over-the-top and totally out of character.
Hey, she's already pretty familiar with regeneration and different Doctors, she's even been in an adventure with three different Doctors (Matt Smith, David Tennant and John Hurt), in The Day of the Doctor 50th Anniversary Special, and having already met (and liked) an older Doctor (Hurt), her overly-negative reaction to Capaldi because he is "old" is totally unrealistic and out-of-character.
The shame is that Jenna Coleman is a pretty good actress, and Clara, as a companion, is quite likeable. But Steven Moffat didn't come up with the writing goods for her on this occasion, writing her not as herself, but reducing her to some kind of a one-dimensional cypher, a dig at and representation of fans who can't handle regeneration and the replacing of "their" Doctor with a new one. In her defense, I have to say that this is NOT Jenna Coleman's fault, and the young lady can only deal with the scripts she's been given.
Another major plot flaw was actually one of the nicest scenes in the entire episode, the phone call from the Matt Smith Doctor on Trenzalore to Clara, said call obviously being made before or during the events of The Time of the Doctor. It's all very poignant, heart-tugging and well-acted, until you actually stop and think about it, and you realize that it simply couldn't have happened. It was was a major continuity flub and sloppy writing by Moffat. Nice, emotional, tear-jerking writing, but sloppy and wrong, wrong, WRONG.
MattDoc says to Clara that "the time is getting close", and "it's going to be a real whopper" (he's obviously referring to an "upcoming regeneration"). But remember back to what happened at the end of The Time of the Doctor. The whole crux of the story was that the Doctor was coming to the end of his final incarnation. There weren't going to BE any more regenerations. As far as MattDoc was concerned, for pretty much the ENTIRE episode, he'd run out of regenerations, and he was going to die. That was the whole point of the story. Which is what would've happened if Clara hadn't begged the Time Lords to save him, as they did right at the end, by popping up at the last minute and giving the Doctor a new cycle of regenerations. So Matt Smith's Doctor couldn't have made that phone call. He didn't know he was going to regenerate before it actually happened. Very sloppy continuity mess-up on Moffat's part.
Those were my two main gripes, and the rest I can live with. However, one thing I couldn't complain about was Peter Capaldi's performance as the new Doctor, which was excellent, top notch. I think he's going to be an excellent Doctor. For that reason alone, I'll give Deep Breath a B instead of the C that I thought the fairly average story by itself actually deserved.