A remarkable thing happened when Patrick Troughton took over the leading role on Doctor Who - the show changed from 'occasionally good', to 'often great'. I can't put that entirely down to the character, because it seemed to me a result of the writing and the types of stories that they wanted to tell. Half of William Hartnell's run were historical type stories, where our heroes would be temporarily trapped in the past. The other half would be futuristic or science-fiction based. After the Doctor's regeneration, this changes - there is only one 'pure history' story (probably a leftover script) and that's it, not a single other. Every story from then on is either set in the future, or set in the present day, or has some sort of alien threat or science-fiction element to it. Oh, and monsters. Lots of monsters. The start of 'The War Games' is even more stark, then, since it seems like the first historical episode in three years, and even then it turns out it isn't!
Basically, I want to point out that the improvements made (and they were massive improvements - I was tempted to give up during some of Hartnell's more boring episodes) are not purely down to the change in actor, nor just the stories, but both of these things combined. I can appreciate the difficulties Hartnell had, but also the writers, in pinning down exactly what sort of show they wanted Doctor Who to be. Arguably, that's something that is still happening to this day.
There were some other series mainstays introduced during Troughton's run. For one thing, the title sequence changed (finally!) and introduced 'the face'. We also got the first use of the Sonic Screwdriver, first use of the alias John Smith, and first appearance of the Earth unit... er... UNIT. And, of course, there was a new Doctor himself.
Thoughts on the Second Doctor:
Immediately after regenerating into his new appearance, it's clear Troughton's portrayal of the character is markedly different. It's a confident character, more on top things, more capable. He's still a little self-involved and weird, but ultimately compassionate. He has a few quirks of his own (a recorder!) and manages to make the character something new.
Troughton gives a thoroughly consistent performance. Whether he's shouting panic-stricken commands to people or engaging in more solemn discussion about the wonders of time/space travel, so long as the writing it good, he's always enjoyable to watch, and he becomes what the show needed him to be - a strong leading character. Nonetheless, he's almost always helped out by the supporting cast, and at times even Jamie has to help set him back on the straight path.
Episode Highlights:
It's difficult to pick out the best episodes of Troughton's run. Firstly, because the quality is more consistent, so few stand out as remarkable against the rest. Secondly, because so many of these serials are incomplete or missing that I may favour a completed serial over a reconstructed one, despite its quality. That said, I have managed to choose what I think are the best examples, which I now list below.
The Power of the Daleks (6 parts, all missing)
The Tomb of the Cybermen (4 parts, all complete)
The Web of Fear (6 parts, only part 1 complete)
The Invasion (8 parts, 2 missing but animated)
The War Games (10 parts, all complete)