GrafX2 is an open-source paint program inspired by Deluxe Paint:

https://code.google.com/archive/p/grafx2/

It's harder than it should be to find the latest code, I don't know what the main developer is playing at, would seemingly rather comment on clone repositories than fix the project homepage:

https://github.com/xyproto/grafx2/issues/561

I've found http://www.aseprite.org/ to be a reasonable alternative to DPaint. The UI is quite different, but on the other hand it has lots of more "modern" features, like layers. Source code available under GPLv2, but you can also buy binaries.

Evilpixie (http://evilpixie.scumways.com/) is supposed to be a sort of modern update of Deluxe Paint. It's a one-man free project, so it probably doesn't have a lot of ultra-modern features, but it might suit you.

I've no direct experience, but I've heard it said that "pro motion" is a successor of sorts to Deluxe Paint: http://www.cosmigo.com/promotion/index.php