Download the latest Chunky Launcher from here and start it.
If the launcher doesn’t open with a double click make sure you have Java 8 installed first (get it from here). Make sure that you use the 64-bit version of Java if you use a 64-bit operating system.
Otherwise start it from the console, e.g.
java -jar path/to/ChunkyLauncher.jar --launcher
If you want to get the latest snapshot, check the Download snapshots checkbox. Then press Check for update to get the latest version of Chunky.
If there is a new version, a new window will open. Click on Update to new version to download the new version.
If something goes wrong while updating, make sure that you are using the latest version of the launcher.
After updating, click on Launch Chunky to start Chunky. If there was no update available, just click on Launch in the main window of the launcher.
This is the most important step. Enjoy using Chunky to render your 1.14 and 1.15 worlds!