The Chase is a guilty pleasure of mine, as it is my favorite First Doctor Dalek story. Its just so entertaining, and that is why I watch Dr Who in the first place. That said, there are some filler aspects in the story and some overall dumb things that happen. The first thing I did was cut all of that business with the space-time machine, because its clearly filler and a bit boring in my opinion. Instead, we start with the TARDIS landing, followed by Ian and Vicki separating from The Doctor and Barbara. Cut to the TARDIS and we see the Daleks first appear on the Doctorś screen. That makes up the two minute intro before the title sequence.
Once the titles are done, we cut to Ian and Vicki discovering the trapdoor, followed by the continuation of the Dalek scene, the second Ian and Vicki scene and the sandstorm that the Doctor and Barbara are succombed to.
This is where we have to bridge the gap between Part 1 and Part 2. In the original, a Dalek comes out of the sand as the cliffhanger, and it is very awkward. I was able to remove the awkward grunts the Dalek makes, but eventually I realized I could remove the shot entirely, instead replacing it with a shot from Part 2 of about 5 Daleks spread across the sand.
Part 2 was the worst to edit. There were very little scenes I could cut, and while sometimes that is a good thing, other times, it isn't. This part is so boring, I got it cut down as much as possible, going from 25 minutes to 17.
Next, after a scene in the TARDIS, we cut to the pirate ship. I wanted to remove the trip to America, because the accents are terrible, so I did, and easily. After some minor abridgements to the pirate ship section, we land on the Mechanoid planet. The Frankenstein/Dracula section was removed for obvious reasons, and it is implied that Vicki left the TARDIS for one reason or another, which is why she is on the Dalek ship.
A few important moments from Part 4 are included, before we cut to Part 5, which is the same way. The Doctor's clone is removed because the actor playing the clone looks nothing like the Doctor, and by added a Dalek scene from Part 3 between prelude and aftermath to that section, it is an easy removal. Part 6 has no useless moments, and ends up like Part 2, just cutting a few redundant lines and such.