When you start the launcher it will ask to upgrade itself if newer version is found. Click Yes and it open browser pointing into hosting.zaonce.net/elite/Client-Installer.exe link. Download and install this "Client-Installer.exe" to proceed.
When setup asks install directory, it should offer the existing one you currently have. Use this, so you don't lose your existing log and other config files by selecting new / wrong directory.
When setup is done, start the launcher.
You most likely have to type in your elite dangerous login info now. Click Log-in button and type username/password. Wait for email you get from frontier and type in verification code from it.
If you have elite upgrades available, just click the Upgrade button and wait patiently.
Depending on the popularity of elite upgrade release (are there other than super popular? haha) it might take a while to download and upgrade.
When its done, Play button appears.
Launcher offers different game version options to install, assuming you have access to these. Public or so called live version and possible beta if currently available.