There are a number of pitfalls that can prevent SKSE and the game from working correctly so I suggest using a vanilla profile with no mods installed to work through these:
Do not install Skyrim or Mod Organizer into C:\Program Files, C:\Program Files (x86) or any other UAC protected folder. Note that Windows 10 adds extra layers of protection on these folders that may prevent Mod Organizer's virtual file system from working.
Do not install Mod Organizer into the Steam or Skyrim folder because the virtual file system will not work correctly.
Using File Explorer, right click on ModOrganizer.exe and click Properties. If you see an Unblock button or checkbox, you need to click Unblock and click OK. If you are on Windows 10, you may need to repeat this for every DLL in the Mod Organizer folder and its subfolders. The recommended solution is to download Mod Organizer from Nexus, immediately unblock the file you downloaded, and then install or extract the files to the appropriate folder.
Run SkyrimSE.exe from Mod Organizer to verify you can run the base game without any issues.
Ensure you are using SKSE 2.0.16 and Skyrim SE 1.5.80. You can find the version number by right clicking SkyrimSE.exe, go to the details tab, and look at Product Version. If this is not 1.5.80, you'll need to run Steam and let it update Skyrim SE to the latest version.
In your vanilla profile (without any other mods installed and with a clean Data folder), install the SKSE scripts and try to run skse_loader.exe from Mod Organizer. You can do this my clicking install mod and select SKSE in the folder you downloaded it to. If you see a red "No game data", right click on the Data folder and click Set data folder.
When you install SKSE, you should copy only skse64_1_5_80.dll, skse64_loader.exe and skse64_steam_loader.dll to the Skyrim Special Edition folder (the folder containing SkyrimSE.exe). All of the scripts should be installed as a mod in Mod Organizer using the steps above.