IMO Postman is great for trying new API queries/testing the response you get from any given query.
Create a new query (duh)
Set URL to "https://chapi.tauri.hu/apiIndex.php"
Under params, add a new key as "apikey" with the value being your api key
(this step is probably not necessary) Under headers, add a key as "Content-Type" with the value "application/json"
Under body, set the type to "raw" (default should be "none")
A combobox should pop up to the right, with the default value being "Text" (unless you set the Content-Type header previously), set that to "JSON" (probably not necessary but it enables syntax highlighting with is nice)
Write your request in the textbox below in JSON format (example below)
Click the send button (if you use https, you'll probably get an invalid certificate warning, there should be an option next to the warning to disable cert validation)
IIRC Burning Blade has no API support, try a different realm.