You could do the following to make the Skyrim Launcher launch with SKSE:
Go into your Skyrim folder.
Rename TESV.exe (the application that launches Skyrim) to Skyrim.exe, and skse_loader.exe to TESV.exe. This "tricks" the launcher into thinking that it is loading Skyrim where it is, in fact, loading the SKSE loader that then loads Skyrim.
Go into your Data folder, then the skse folder. If it doesn't exist, create it.
Make a file named skse.ini using notepad, and in it, put the following down:
[Loader]
RuntimeName = Skyrim.exe
- Save the file, and run Skyrim using the Skyrim Launcher.
Side note: If Nexus Mod Manager notifies you that SKSE isn't installed, ignore the warnings and install the mod. This is because NMM checks for SKSE by checking if there is a skse_loader.exe file in the Skyrim folder. And obviously, as you renamed it, it is not there.