15 R3, HP Thunderbolt dock, not working

Even if there's a hardware issue here, if you don't have another system to test the dock with or another Thunderbolt device to test the laptop's Thunderbolt port with, it will be tough to answer which device is the problem.  But before assuming there's a hardware issue, make sure you've updated your system's BIOS, Thunderbolt 3 controller firmware, and Thunderbolt drivers/software.  Most systems and Thunderbolt peripherals on the market as of this writing still have to be "approved" for use before the system will work with them.  This is related to the fact that Thunderbolt allows access to PCI Express, which in turns allows direct access to system memory, and that can be a security risk.  Newer Thunderbolt systems and peripherals support some additional protection mechanisms that manage that risk without allowing per-device approval, but they're not as common yet.  Anyway, when you first connect a Thunderbolt peripheral, the Thunderbolt software should trigger a popup message asking you whether to approve the device once or permanently.  If you're not seeing that popup, it could be because the software isn't installed.  If it's installed and you've updated the other components I mentioned and you're still not seeing it, then you could check your BIOS settings to make sure the system's Thunderbolt capability hasn't been disabled, but that would be unlikely unless you'd done it manually (or you bought the system used, I guess.)  At that point a hardware issue would be possibility, but without being able to test each component with another device, narrowing it down will be trickier.

Ok, thank you for the very detailed answer, that already helps a lot. I tested the dock meanwhile with another laptop, and there it works. I also plugged in another USB3 device at the port of my Alienware, and this works as well, so it's basically the combination of this particular dock and my Alienware. Have to figure out where to get the latest drivers now, is there any chance you can direct me to a site here?

Thanks and regards
Martin

@TiRohn  drivers can be found by going to support.dell.com and searching your model name or Service Tag, then clicking the Drivers tab.  The direct link to that page for the Alienware 15 R3 is here.  You may also be able to install Dell Update (or maybe Alienware Update), which would check for updates that apply to your specific system configuration and periodically check so that you get notified when updates are available.