These are my rankings of Doctor Who stories from the First Doctor period. I have categorized them into five categories, rather than finding a linear ranking order for each story. For links to the entire series, see here.
10. The Dalek Invasion of Earth: One of the best First Doctor stories. It's the second appearance of the Daleks, and given the original naming conventions (where individual episodes were named, not overall serials, as became standard practice later in the show) you wouldn't have gotten the presence of the Daleks spoiled by the title until the end of the first episode. The TARDIS crew ends up in 22nd Century London, where the city has been devastated, with very few people in sight, all of them acting in a robotic manner. When they discover the first Dalek they come across, it's a bit of a shock, because they'd only met the Daleks on their home planet in their first appearance. Despite a ridiculous sci-fi premise for why the Daleks have invaded Earth, this story works incredibly well, which certainly isn't true of all the Terry Nation Dalek stories in this period. I don't think it's his best. That honor goes to The Daleks' Master Plan. But this is among the truly classic stories of the First Doctor period.
21. The Daleks' Master Plan: This is by far my favorite First Doctor story. A full dozen episodes (a baker's dozen, if you count the prologue episode Mission to the Unknown, which came two stories before but was really part of this story). Unfortunately, only three episodes survive, so you either have to listen to the soundtracks for the rest or watch the fan-created reconstructions based on the large number of set photos that exist and the existing soundtracks. But it's worth it. The stakes are higher than any previous Dalek story, and it has better good science fiction concepts than many of the other non-historical earlier episodes. We get to see a future Earth empire with a military that knows all about the Daleks and is trained to fight them, including two noteworthy characters, a brother and sister played by Nicholas Courtney, who later went on to play Brigadier Lethbridge-Stewart, and Jean Marsh as Sara Kingdom, one of my favorite companions over the entire run of the origianl series. Marsh also had earlier played Princess Joanna in The Crusade and much later returned to play Morgana in the Seventh Doctor story Battlefield, which was also the final appearance of Brigadier Lethbridge-Stewart in the Doctor Who show. This was the only story featuring Sara Kingdom, unfortunately, but she's present for something like eight or nine episodes of it. Terry Nation wrote episodes 1-5 and 7. Unfortunately, the seventh was a Christmas episode that has nothing to do with the rest of the story, which is its only real low point. By that point in the story, we're reliving The Chase, where the Doctor, The Meddling Monk (from The Time Meddler), and the Daleks are running around through time, and it slows down a bit, but those parts are a little better than the middle episodes of The Chase in my view. But the first half of this story and the last two or three episodes are as enjoyable as the First Doctor gets, even with reconstructions of the episodes.
23. The Ark: This is one of the better "future of humans" stories of the First Doctor. The TARDIS appears on a human ship in the future, and there's another intelligent species serving humans as slaves, in effect, although from all appearances it's consensual, and the humans are unaware of the full intelligence of these beings. Halfway through, the TARDIS crew has resolved their original problem keeping them there, and they reappear in the same spot but much further in the future. Since this is a time when the Doctor had no control at all over where the TARDIS ends up, that seems remarkably odd. Then they discover that a revolution has occurred, and the other species has turned the tables on their human masters. Instead of being victims that we feel sorry for, they are now the villains. This was a nice nod to the common phenomenon in human history of the victims gaining control and becoming just as bad oppressors as those who had oppressed them. We also get to see an invisible (i.e. money-saving) but very powerful alien race that reminded me much of the sort of thing you might see on the original series of Star Trek, which was being made around the same time period as this episode. This episode didn't win me over to new companion Dodo. But it has some funny moments between her and the Doctor, where her slang expressions (that are entirely commonplace now, to a point where it shocked me that anyone wouldn't be used to it) give us a glimpse of the First Doctor's cantankerous nature in his complaints that she's not speaking English (which I should note is her first language and not his). And this is one of the few First Doctor stories that I'd gladly show to someone who wanted to see a good example of what the best of his period was like.
Very Enjoyable Stories
2. The Daleks (AKA The Mutants, not to be confused with a later Third Doctor story): This is the serial that gave the show its initial success. It drags a bit about 3/4 of the way through, but overall this is a great introduction to the Daleks. As with most of Terry Nation's Doctor Who stories, there are deeper themes to the story than just an action/adventure romp. In contrast to some of the emphasis of later Doctor Who stories (including some of Nation's own), here we see the Doctor encouraging pacifists to take up arms to destroy a menace that would otherwise end up destroying them. This is one of the best First Doctor stories.
17. The Time Meddler: In this story we get the introduction of our first Time Lord character (not that we have that name yet) besides the Doctor and Susan, and we even get to see his TARDIS, both inside and out. His chameleon circuit works, so we see a TARDIS properly disguised. The Meddling Monk returns as well in the Daleks' Master Plan, so he's also a recurring villain. A renegade Time Lord seeking to change history for some unclear profit motive (or perhaps for some higher good, but in any case the Doctor disapproves), the Meddling Monk has set himself up at a monastery, where he's pretending a whole group of monks are present by using future technology (including a phonograph with recordings of medieval-style chant) to give the appearance of a larger population of monks (as well as to make his stay more comfortable with appliances such as a toaster). The Doctor and his companions eventually figure out what's going on, and the Doctor manages to show some know-how when it comes to how a TARDIS works by sabotaging the Monk's TARDIS (which unfortunately never manages to help him get his TARDIS working properly again so he can actually control where it goes, not until the Time Lords help him later on during the Third Doctor period). This is the first time we see a historical setting with something non-historical worked in, a formula that the show eventually uses almost exclusively for stories taking place in the Earth's past, but we still have another season or so of purely historical episodes to go before that becomes standard. It's the first time also for the new lineup of the Doctor, Vicki, and Steven. It has some moments of lagging, as historical episodes tend to do, and it's the first historical episode with discussion of the real possibility of history-changing (see The Space Museum for the first instance of this, however, although this doesn't have the complete incoherence of that story). That is a disappointment from the perspective of metaphysics, but the unique elements of this story more than make up for it.
27. The War Machines: This is one of my favorite. If it weren't for the musical companions, it would be in the top category. The adventure starts with the Doctor and Dodo arriving in Dodo's own time period (roughly the time the episode aired). She's in the first episode and maybe part of the second. She never even appears to say goodbye to the Doctor. It introduced Ben and Polly, but Polly is brainwashed for most of the episode, so we don't get to see her in her right mind very much. And much of the episode Ben hasn't really connected with the Doctor. So it's not really the usual Doctor and his companion (or companions) sort of piece. That being said, this was a great introduction to what became a much more standard format for the Second Doctor period, where the Doctor (and in the other cases his companions) is in the time period when the show was being made, the mid-late 1960s, fighting off some menace threatening the time period of the viewers of the show. In this case, it's an artificial intelligence that, in a rare case, seems to have nothing to do with aliens, but you do get some rather rudimentary-looking robot threats (in keeping with the era they couldn't have them be too sci-fi looking). The Doctor uses logical paradoxes to undo the machine, as he does in several other stories (The Green Death, Death to the Daleks, and Shada come to mind). I do tend to like Ben and Polly, but we don't see a lot of Polly in this one. There's a nice scene at the end where the Doctor thinks he's all alone for the first time since the show began, but he ends up getting surprised with some unintended stowaways at the end, leading into the next season (and his final two stories).
29. The Tenth Planet: This is the introduction of the Cybermen and the last story for the First Doctor, so there's particular significance to it, but it doesn't work as well as I'd like. The Second Doctor Cybermen stories are much better. They look like they're wearing cloth outfits instead of metal. It's hard to hear what they're saying sometimes. The Doctor is showing his age, and several of his scenes had to be given to Ben or Polly. (Both Hartnell and the character are dying of old age at this point.) At the end, after defeating the Cybermen, he just collapses and dies, only to be regenerated into the Second Doctor. They don't explain the regenaration all that well, and the final episode is missing (although there are copies of the regeneration scene that have been released on DVD and online). Fortunately, this is one of the missing episodes that have now been animated. Still, this is a decent base-under-siege story, a template that becomes much more common with the Second Doctor, and as the introduction to the Cybermen and the final First Doctor story, it's certainly one to see.