If the rain is coming under your walls then in time the bottom of your walls will rot. Adding dirt against the wall outside or inside will help the walls rot sooner.
If the rain is entering through the crack in the concrete clean the crack and fill with high expansion foam.
As you suggested a gutter to redirect the rain away from the coop. Discharge in a rain barrel for watering the chickens? Barrel overflow as far away form the coop as practical.
Borrow, rent or purchase four hydraulic jacks (3 or 5 ton should do it). Dig under each coop corner to place a jack with some blocking. Slowly raise each corner a little at a time, going from corner to corner to corner to corner to corner...until you have the entire coop raised high enough to slide 4" x 8" x 16" solid cement construction block under the walls of the coop. Lower the coop. Now you have the coop 4" higher then what it was. Fill the inside of the coop with your choice of bedding (sand would be my choice). Outside use dirt from the top of the new blocks and slope away from the coop. You could use just one hydraulic jack, raise, block with scrap lumber, remove jack to new location and repeat. May sound harder then what it actual would be.