What kind of soil do you have now? Is it clay which is impervious and doesn't drain or is it sand that drains really well? Most of us have something in between. Does water drain away from it now or does water drain to it and stand? How big will the run be, in feet.
If it is on sand, it's probably not a big issue. If water drains away from it you are in good shape but will have some issues on clay. In a low spot with clay it gets harder. If you are in a soil that holds water and are in a low spot, or if you dig a hole in that and fill it with sand, it won't drain. The water has no place to go.
I'll include this link now. It has a lot of good ideas in it.
https://www.backyardchickens.com/articles/how-to-fix-a-muddy-run-chicken-coop.47807/
Putting a roof over it can help, but rain still blows in from the side. My run is on a high spot so it drains reasonably well but it is dirt. It is covered. When the weather sets in wet it still gets really muddy. I've poured pea gravel in strategic spots to build it up so I can feed and water out there but mostly live with it. The chickens have drier spots to go.
Some people spread bedding on something like that. we all have our favorite bedding. In some circumstances some are better than others but many can be made to work. Some people can just add bedding and leave it. Others may need to remove it and replace it as it gets soaked.
If your area is low and holds water, you may have to work a lot harder, especially if you don't fix it as you build it. Read Pat's article, it can really help. If your area isn't too big and there is a nearby lower spot water can drain to, digging a trench between your run and that spot and filling it with sand or gravel can move a lot of water. When I built my coop I hauled in dirt and raised the floor a few inches so it is higher than the surrounding area, but rain does not blow in from the sides in my coop. Still, it can help with a run.
Without knowing your situation it's hard to get too specific. You may be OK with natural dirt, but probably not. Sand may work great or may be a bad choice. Some people make walkways by laying pallets or something in there so they can stay above the mud. Some people use bedding: wood shavings, wood chips, hay, straw, dried leaves, who knows what. Some people turn their run (or coop) into a compost pile, tossing in dried leaves, grass trimmings, various bedding, kitchen wastes, and garden wastes. Some people use sand and scoop the poop every day, like a cat litter box. That doesn't help mud but can make it nicer to work in there.
There is not just one way to go about this, you have all kinds of options. We are all unique, something that works for me might not work for you. Good luck and welcome to the adventure.