What is the proper way to run drain tile for downspouts?

Not sure if you're talking about a permanent draining system around a basement foundation that feeds to an inside sump pump or are you wanting to move downspout water away from the foundation of your house but have it underground where it's not seen (like when you use above-the-ground downspout extensions that are visible and potential tripping hazards). If it's the latter, consider installing a French drain system: https://www.hgtv.com/remodel/outdoors/how-to-install-french-drains

Are you just going to have the downspouts drain into it? If so, I’ve seen PVC pipe systems made for this application.

I would have at least a 2% slope, the entire run. Personally, to reduce icing up, I’d go 4%. I would start at a depth of 6” minimum.

Many cities have Codes on the discharge of the drain systems out onto the curb, so check that first.

https://www.youtube.com/watch?v=FwKPpU-8oes

https://www.youtube.com/watch?v=ByVKaApxd8A

https://www.youtube.com/watch?v=rT6ZUWXsYgQ

I dug a trench at a couple of my downspout locations and put in 10' of drainage pipe with the downspout directed into the pipe. The pipe goes down the slope of the side yard and into a pit filled with rocks. The water dissipates into the ground there. We only get 7" of rain in eastern WA, so we don't need to worry too much about it. In fact most houses don't even have gutters. I wanted the water away from the house on the side so it doesn't wash out the flower bed.

Hi Scott, hope this helps you out,

https://www.youtube.com/watch?v=1djQ5rmCGo0