I am customising a premium WP theme to suit my site better, but I can't quite figure out what to do with PHP files that I customise. Do they just go into the root of the child theme folder? Or do they get put into a similar folder hierarchy as in the original theme? How do I make sure the child theme PHP file is being used instead of the parent theme PHP file (e.g. header.php)?
To take your questions in no particular order:
Any template files you include in a child theme will be used instead of an identically-named parent version automatically. (This doesn't apply to style.css or functions.php, but header.php etc are all covered.)
WP will use the template hierarchy to figure out which templates are needed based on the query in question -- if the theme is well-constructed, you should be fine putting them in the root of your folder without replicating the original directory structure.
BUT, if the parent theme was coded in non-standard ways, and in particular if it tries to include files directly rather than using wp functions (like get_header(), get_footer(), get_template_part() etc.), you may have to get your hands a little dirtier. Good premium themes will often have documentation on how to build child themes for their own systems, especially if they depend a lot on non-wp hooks to get things done... but of course, you may not know until you start playing around how the documentation is on yours. Good luck!
Couple of things to add
So How do I make Sure Child theme file is used: Well, if you do everything right child theme templates already getting used. But still if you want to check you can just add a html comment on the template file and check the source. I do it sometimes when things doesn't work for me also I add the php file name to the html comment so i know which file includes/requires from where.