The most fun way to run Macpaint today is by using a computer it was actually designed for, an old Mac that you can find on eBay or at a thrift store.

It will run perfectly on any classic Mac OS (system 1-9). If you are trying to run it under OS 8 or 9, you will probably need to switch your color depth to Black and White. Running on a 68k Mac (pre-PowerPC) with system 6 or 7 will give you reasonable usability and plenty of authenticity, but it will definitely run on any OS through 9.

The more practical way is to run a 68k or PowerPC emulator on a modern Mac/Windows/Linux PC. Basilisk II and Sheepshaver work well enough that they are worth trying if you want a full-featured emulator to run systems up through 9.0.4. Note that PowerPC ROMs are trickier to come by, as are OS 8 and 9, so you may want to try Mini vMac, a great Mac Plus emulator that is easy to set up and will be fine for Macpaint.

 
Those with no experience with pre-OS X Macs will definitely want to start with Mini vMac. Using at least System 7.1 with Basilisk II or Sheepshaver makes sense, and for all intents and purposes this requires a new large HFS-formatted (not HFS+) disk image, which is non-trivial to create these days. Freely available floppy images are more than sufficient for Mini vMac.

If any terms here are unfamiliar to you, start with Mini vMac. It's very easy to use and works well.

*Emaculation is a site that may be very helpful to you
*Here are some convenient prepackaged System 7 and Macpaint disk images and instructions for vMac (instructions and supplied vMac version are for Mac OS X, but it's similar on any system)
*Many useful disk images, including Macpaint 1.5
*Download System 6.0.x, 7.0.1 or 7.5.3 from Apple legally
*Users of OS X 10.6 or higher will need FuseHFS to write to HFS standard volumes (disk images)

NOTE: it is illegal to copy and distribute Mac ROMs, which are required to run most emulators other than vMac. If you're going to do this, make sure to rationalize it somehow. Some of the Apple OSes you may want to use are also not freely available from them, so do be careful and ethical online.