"What's the best way to do this?"

The best way to do this is to not develop on your production server. For an (arguably, better) alternative, check out Why & How-To Set Up a WordPress Local-Development Environment With Vagrant.

Otherwise, check out sections 2 & 3 of Giving WordPress Its Own Directory | Codex.

Pablo, thanks I do develop locally using either MAMP or XAMPP. I probably shouldn't have said dev sandboxes. They will be more like dev presentation boxes. I got DesktopServer recently and it's pretty sweet. I can share out my IP and let clients view the sites, but I can only share one at a time.

I am often working on multiple sites at once and I want to place to upload them to where my clients can see them before we migrate to their live server.

Thanks for the link to the Wordpress codex. That looks like that would work.

What do you do for sharing sites with clients to get them cleaned up before migration and launch?

"What do you do for sharing sites with clients to get them cleaned up before migration and launch?"

Exactly what you're trying to accomplish.

1.) Create a subdomain, e.g. client_name.mydomain.com;

2.) In DigitalOcean's DNS Manager, create an A record (pointed to a 2nd, $5 droplet);

3.) Install WordPress in a subdirectory named after the client, e.g. /srv/www/client_name/.

I pretty much follow the process in How To Set Up Multiple WordPress Sites on a Single Ubuntu VPS (but substitute Nginx in lieu of Apache); although, I did (by accident) come across a new article recently: How To Configure Single and Multiple WordPress Site Settings with Nginx (but haven't gotten a chance to work through it, yet).

(RANT: I really wish DigitalOcean would make it easier to peruse newly published articles -- as opposed of having to go chase them down on Social Media! Okay, I'm done.)