I think this is one of those basic questions that has a really complex answer. Possibly because every writer you ask is going to have a different answer specifically. (See Susan Dennard's method or this quote from Holly Black.) I do feel, though, that many will have the same idea generally.
There are tons of ways to come up with characters--and even how to flesh them out later. You can look at archetypes, have an idea of the backstory, or maybe where their arc ends. Ideas may come to you from a picture on Pinterest or a random song. But building out a well-rounded character is more than just having a compelling backstory.
Each character should have their own goals, dreams, and "faults". Really relatable characters have a reason for every decision they make, and that comes from experiences they've had before they even step foot on the page of a book.
I think that's the most important thing to writing great characters--and ultimately great stories--is remembering characters are people too. Even though they're fictional, they don't just show up on the page and disappear once you close the book. These characters had a childhood. They had experiences that shaped how they think in the book you're currently reading. And we need to have a deep understanding of the why.
First, inspiration has to come from somewhere. So, if you're looking for where to find it, here are just a few starting places:
Pinterest
Music
Go for a walk
People watch
Sketch something random
Look up random names
Search story prompts
Art museums
Once you have an idea of the character you want to build out, start considering what they want, and also the belief you want to demolish in them. This is going to be the thing that drives your story forward. What was the key moment in their past when that belief was formed? How did it shape them moving forward? And what's the external factor that is going to force them to confront that belief--and ultimately change because of it?
Personally, I feel that emotion is the thing that draws readers in. Finding stories that make us question how we would react in certain situations without actually having to go through them ourselves.
Knowing the belief the character will need to overcome will help us to understand where they start and also where they are by THE END.
These are the foundations from which other fun character elements will be established. Things like their favourite drink, if they like dogs or cats more, and how they feel about rainy days. But the truth is, as much as these things may help us to understand our character, they only matter if they come up in the story.
There are probably going to be things you find out about your characters as you write. It's totally okay to start with ideas and flesh them out as you go. Some things may even change from what you originally planned. Go with the flow.
But, hopefully, this will help to kick-start those character gears! I'm currently reading Story Genius by Lisa Cron which I highly recommend for establishing characters who drive the story forward (and it also goes deeper into the ideas I touch on here).
Do you have an established method for writing characters?
Happy writing,
Erin