Character Development: Unfortunately, there is none regarding Holmes's return to life — he gets nostalgic from time to time and he refuses to accept robo!Watson at first, but this is as far as it goes. One would think he'd have someissues after having died and then returning to life in a completely different century.
However, as mentioned above, he goes from rejecting Watson to embracing him wholeheartedly in the span of two episodes.
Plus, his relationship with Beth develops subtly throughout the series.
Destructive Saviour: Grayson reads a laundry List of Transgressions inflicted in the process of catching Fenwick in the first episode, and it's implied it isn't the first time her enthusiasm for catching criminals has come at a price.
Drives Like Crazy: Demonstrated a fair number of times over the course of the show, enough to be gently snarked on the topic by Holmes.
Hot-Blooded: Lestrade is quite unrestrained in her emotions, as a contrast to the calm and genteel Holmes.
Inspector Lestrade: Literally in name, and like her ancestor she starts off making some hot-headed presumptions in cases as to who is guilty of what. That tendency smooths out over the course of the show.
The Knights Who Say Squee: She is both an officer of the law and a devoted admirer of Holmes' talent for outwitting criminals. Enough to suggest revivification to a highly skeptical Chief Grayson.
Legacy Character: Of a sort. She is directly related to the original Inspector Lestrade and assumes a similar role to the first as Holmes' point of contact with Scotland Yard and actual figure of authority in a case. Her relationship is much less vitriolic than the original relationship between Holmes and Lestrade.
Trigger Happy: Lestrade draws her ionizer at the earliest opportunity in many cases. Fortunately she tends to holds her fire if innocents are in the way. When she only has to worry about a suspect, she doesn't hesitate to fire.
Turn In Your Badge: Threatens to do this to Grayson and take her story of a revivified Sherlock Holmes to the news to make New Scotland Yard look like they were shirking their duty to help the public by ignoring potential assistance.
The Watson: Goes back and forth on this; occasionally he asks questions to give Holmes a chance to reveal what he has discovered, but will also bounce his own deductions and inferences off Holmes.
What Measure Is a Non-Human?: "The Fall And Rise of Sherlock Holmes" and "The Crime Machine" let Holmes explore his relationship with Watson. "The Five Orange Pips" revolves around this trope.
Chief Inspector Grayson
Arbitrary Skepticism: No matter how many times he sees Sherlock Holmes crack a case, pick up on facts no one else noticed, and solve crimes, he constantly refuses to believe that Holmes is any help at all.
By-the-Book Cop/Lawful Stupid: Grayson straddles the line of this due to his unwillingness to deviate from police procedure for even a moment.
Fat Idiot: His portrayal, albeit probably not intentionally so by the writers. Comes across as this, as the result of his combination of husky build (with an obvious paunch) and his incompetence in the course of fighting crime, including once being Moriarty's Unwitting Pawn to provide an alibi for a crime. He was also introduced getting an elasto-mask of an old woman stuck on his head.
Even Evil Has Standards: According to Moriarty in "Five Orange Pips," he doesn't want people getting hurt as he takes over the world. To be fair, he generally sticks to this rule, with the exception of "Baskerville" early on in the game.
Evil Genius: Positively brilliant, considered Holmes' equal, and as dedicated to committing crimes as Holmes is to solving them.
The Starscream: He accomplishes this offscreen. Originally cloned to be Fenwick's Dragon, he upstages Fenwick as the true criminal mastermind.
Took a Level in Badass: This is a man who enjoys getting his hands dirty and enjoys sparring verbally and physically with Holmes, quite unlike the reclusive intellectual of the Canon.
It does make sense, though, as this Moriarty is a clone of the original created by Fenwick, who was supposed to serve as his Dragon. Presumably, there was some manipulation involved to make him more suited to the task.