I do a lot of talking to myself.
Character and story tend to be very closely linked for me. I can't just make up a plot and then slot characters into it - I need to create them both side by side, because they feed off each other - so I talk myself through my stories a lot, by telling it to myself first. If something sounds off - say, the dialogue is stiff - or if a scene just isn't working because it doesn't fit into the flow of the story, I know that before I have to sit down and write/draw it all.
I also do a lot of reading on things that are related to the story - both as part of the focused research stage, and as part of what made the inspiration spark in the first place. And it isn't always the most obvious things, either.
Say I'm doing a comic about a space colony - well, then, I probably read stuff about people surviving in hostile environments, but also stuff about our world's colonial past, to see how colonies were organised and operated, and what political and social impact they had/have on the people involved. I probably read about historical examples of different cultures and religions meeting and combining into something new, with aspects of the original.
For Grassblades, a lot of my research has been visual, looking for a visual atmosphere that fits the tone of what I'm trying to tell, but I've also read up on ghost stories, phantom sensations in amputated limbs and selective mutism.
And all the while, I talk. I tell myself parts of the story over and over again, trying out new things, scrapping what doesn't work, and sometimes re-telling the same scene - or the same line of dialogue - over and over with a different emphasis every time, to see which one I should go for.
One thing that helps me out a lot is once I've gotten a bunch of my ducks in a row - the protagonists and antagonists, some general idea of the overall story, some solid worldbuilding for them all to stand on - is to examine what I have very closely, and try to figure out what it's really about. To use Grassblades as an example again; on the surface, it's the story of a very grumpy man with some magical anger-issues being forced to take care of a child that isn't is, as he searches for some way to stop this weird magical disease thing he's got. But beneath that, it's also a story about recovering from trauma, and re-learning what it is to be human (in the emotional sense; I promise my protagonist isn't secretly an alien) - it's a story about trust, about parenthood, about making up for our mistakes, etc., etc.
I'm not setting out to write some Big Message (tm), but I find it's useful to have those themes in mind when I plot out the story, because it helps me set the tone and frame the scenes in a way that ties into this, and it also helps me come up with secondary characters and subplots that reinforce and/or mirror the protagonist's experience with those themes.