I take it that your emphasis is on inexpensive lighting kits. Good news: for interviews, you can do very well indeed with a relatively small outlay.
Key: a Chimera (could be “small” or even “extra-small”) with an open face fixture, for which you’ll also need a “speed ring” to go between the fixture and the Chimera;
Fill: a reflector (could be a piece of foamcore);
Background: a Source 4 Jr. to put a pattern on the background.
There are cheaper versions of the Chimera from other suppliers. The background light is important but it not absolutely necessary. eBay is a good source for a used Chimera/light fixture/speed ring, but make sure you get a “video” or “hot light” Chimera instead of the still-photography kind (which can’t stand the heat and are designed for strobes).
Alternative for a key: a fluorescent.
Ultra-cheap key: open face fixtures with diffusion on the front. You can get a very cheap but durable Mole Nooklite on eBay periodically.
Ultra-ultra-cheap key: One of the “best-lit” interviews I ever shot was with a large window providing 100% of the lighting: a very soft, broad, natural look.
No matter what you buy, HOW you use it is much more important. I guarantee that some of the experts on this forum could light an interview better with a crappy light kit than the less-expert could with the most expensive light kit. Search the forum! volunteer to go on a shoot with a great local shooter!
Good luck, take pictures of your lighting setups so you can duplicate the good and avoid the bad, get on with it and start to play.
— BC