What are the bases that you rely on to think that a champion needs an update? and what bases do you rely on to think the updated champion is where you need him to be?

Let's make a wide statement!: Every champion could use an update in one way or another, and this becomes more true as time passes. When Champion Update (then Relaunch) first spun up, we thought that Vayne was the bar that we'd stop at, and nowadays while Vayne isn't horribly offensive, she could use some love (in the art spectrum).
To your question, it drastically varies by champion. Lee Sin is fun, and has a good amount of complexity. While we could update his kit (I'm preeetty sure we're not looking at this any time soon), I think everyone would agree he'd be a much cooler character if he didn't look like he lacked a collarbone (go look at it). Contrasting, Aatrox is a modern character in appearances, but his kit probably isn't in the healthiest area.
Obviously, some champions need updates more than others. Yes: Taric, Poppy, Urgot, Eve, etc. These are generally the highest examples of work needed, and I won't really go into them. Just mentioning them so you don't think I'm ignoring them.
Past this, there's the ones that have high opportunities, but don't require a whole lot of work (or can at least pass with minimal work while we work on the above characters). Singed and Alistar are great examples of this. The visual polish goes a loooong way to improving them. Not the end of the line, but we are just getting work in while we have the additional bandwidth. Anyways. Art stuff is easy to know when they're not fully up to snuff. The other two fields, Creative and Design, are harder to quantify.
Starting with Creative, this can be a character that is literally just a dead end. Not just a character we aren't telling stories with, it could also be a character that's just a complete black spot in terms of execution and fantasy. My favorite person to pick on with this one is Shyvana: The Half(ass)-Dragon. While she has some merit as a character (Farm style jungle kit, her backstory with J4 (J4xShyvana > J4xQuinn)), her execution as a DRAGON is pretty weak, and that bleeds into Art/Design as well. Also, sometimes we just see really awesome opportunities align larger packages, such as Gangplank did with Burning Tides.
With Design, there are several factors at play. A high win-rate, but a perception that they're weak and useless (how do you buff that?). An -extremely- low play rate (niche characters are fine, not every character needs to be Jinx/Ahri). An utter lack of balance levers. Not bringing anything unique to the game (why would you pick X when you could pick Y?). A perception of just... not being fun. People much smarter than I at that generally play that field, but we generally try to hit the "most offensive" characters here.
So yeah. We kinda factor that all in with how much bandwidth we have in an area... and that's roughly how we schedule things.
Our hope is that every character has an audience who loves that character more than everyone else.