Basically, some people started incorporating gloomy aspects to punk aesthetics and music. Gloomy aspects from literature, film, tropes. Some were looking to express similar political sentiments but in a more metaphorical way, I suppose; that's my impression, that there was an added element of artistry/artsy there. Some were looking to add also subjective themes (madness, unlove, etc.). Example (Bauhaus - In the Flat Field).

This gave us post-punk and similar sounds in the very late 70s and mostly early 80s. You probably know some bands that were influential. Example. (The Cure - The Hanging Garden).

Anyway, the mohawk grew longer, blacker. Still teased, often shaved, but creepier. The leyering in clothing also became blacker or creepier (transparent layers, protagonism of the net layers). Theatrical and extravagant outfits emerged, inspired by the decade's fashion too. Example in music video A. (_Alien Sex Fiend - R.I.P.).
Example in music video B. (Specimen - Kiss Kiss Bang Bang)

Famously, the night club called "The Batcave" started reuniting bands and listeners, and in the 80s many countries had similar venues. A little more aggressive or punkier sound persisted with a genre called deathrock. There was now post-punk, deathrock, goth rock, darkwave (inspired by the synth-driven genres of the 80s), etc. Example A. (The Sisters of Mercy - Lucretia my Reflection)
Example B. (The Frozen Autumn - Is Everything Real?).

The subculture was consolidated around these genres, and then the name "Goth" stayed, and then more communication and inspiration... And nowadays social media keeps many things alive through teachings in video and, I guess, text, like this one. Why? In part, because [context] a person that is new to all this world is called a "baby bat" and [/context] most Goths are protective and integrating of baby bats.

And that's it. There was a good deathrock revival in the late 00s and early 10s. Example. (Cinema Strange - I Remember Tendon Water). And today there's an obsession to bring back the original elements but it's all still very different (and often cliched, which I don't like, personally).

Here's some Goth music from 'recent' years. (Lebanon Hanover - Gallowdance).
Another one. (Boy Harsher - Fate).
And another one. (The Cemetery Girlz - Broken Teeth)

More knowledgeable Goths, correct or add as needed...

My understanding is punk came about at least in part of the first wave of ska, not the other way around.

Though I could be mistaken.

Goth-rock started from post-punk, with Bauhaus and Siouxsie and the Banshees following the lead of Joy Division. Not sure what kind of goth-rock you mean that didn't start there.