Rolts I don't know what is causing the issue. I have a lot of experience with Dell computers, going back over 20 years, and I've never seen Linux installation brick a Dell computer. I did a quick internet search and could not find a situation where an E6420 bricked after Linux installation. The reported problems involve wifi, touchpad and other components requiring special drivers to work with some distros, but not a brick.
If the Live session worked, chances are high that something went wrong during the installation process.
But, whatever is wrong, bricked you are, and if you can't get to either the BIOS Menu (F2) or the Boot Menu (F12), I think that you are going to have to do a clean Windows installation before you can do anything else.
The E6420 (circa 2010/2011) should handle a clean installation of Windows 10. If you can, use a Windows computer (yours or a friends) to create the USB installer, because a Windows-created USB will boot the computer without the necessity of using the Boot Menu. Although the computer came with Windows 7, it will automatically license Windows 10.
If you can get Windows 10 installed, use the Dell Support for the E6420 to update BIOS and drivers. Use the laptop's service number (should be on a tag on the computer) to get drivers for your specific computer. If you don't have that, the generic E6420 drivers should be fine.
The E6420's specs are reasonably vanilla (Intel Core i5-2540M, Intel HD Graphics 3000, 320GB SATA HHD, 1366x768 display,Intel Centrino Ultimate-N 6300 802.11a/g/n wifi, Bluetooth 3.0, and so on), so it should work. I checked the Ubuntu compatibility list, but the computer is so old that it is not listed (listings start with the G-series).
A thought: You don't say one way or the other, but have other distros worked on the laptop? If not, consider installing Ubuntu (or a variant like Ubuntu Budgie, Kubuntu or Xubuntu) before trying again with Solus. Ubuntu has excellent hardware compatibility overall, and if you run into a problem with Ubuntu, that will serve as a canary in the coal mine, telling you that something about the E6420 is blocking Linux installation. If the E6420 worked with other Linux distros, then my guess is that something went wrong during installation.
