Note: Chiang Mai is not a cold place, it's 33C in the day and 24C at night. But heat acclimation is a good thing.
My running whilst I was in Chiang Mai consisted of two different types: track work and long runs.
Chiang Mai University has an excellent track and is only about 1km from the Nimman main road. It is open most of the time but it's kinda hit-or-miss when it's free to use (there's probably a schedule somewhere). I found that was it was typically unavailable on weekend nights, due to races, so weekday nights were the best. The track was normally empty up until sunset (~18:30) but soon fills up with about a hundred people. They keep the inner 2 lanes for serious training, lanes 2-4 for joggers and 4-8 for walkers. A small running club marks off the inner lanes on Wednesday nights so it's pretty much just you and them.
Second, there is a huge mountain right next to the city. Doi Suthep towers over the city and has multiple routes up. I set myself the challenge of running up the road to Wat Phra That Doi Suthep, near the top. From the bottom to the temple, it's about 11km of pure climbing. The gradient is so high near the top that you're almost using your hands to push your legs down. I did this by starting at 04:45 and running in the pitch black for around an hour.
There's also the Pilgrim's Trail, which is a lot more direct (and therefore steep). It's around 7km from the bottom to the temple but it can get pretty gnarly when the rain has hit.
I also did a few recovery runs around the city. The Old City walls measure about 6km and it's okay to run along at night (just watch out for the traffic). By taking random sois, you can easily add distance onto the loop and see a bit of the city.
The main thing to remember when running in Chiang Mai is to keep hydrated. The temperature is always high and the humidity makes it hard to stay cool. It's best to take it steady, and build up gradually. Electrolyte drinks/powders are easily found in 7/11. I went to Decathlon, in the south-west corner of the city, to buy a trail-running vest so I could carry enough fluids during my mountain run.