This laptop is several years old. If anything other then the hard drive, memory, WiFi/WWAN card has failed, I would recommend replacing the entire laptop. This mainly has to do with 3 factors, outside of how old the laptop is:
- Dell no longer makes new parts. The majority of parts that remain are used unless you find a NOS part for sale.
- Many parts cost nearly as much as a new laptop.
- Some parts are grossly overpriced today.
The first thing I'd suggest checking is the memory. The first thing you will want to try is reseating it. If this doesn't work and you have multiple modules, test both modules individually to find the defective module. If you find both memory modules are bad, you will need to pick up a new set of memory for the laptop to rule this out entirely. The guide to access the memory can be found here.
If this doesn't fix it, check out the hard drive. Try booting the laptop without the hard drive installed and see if you get any better results. If your laptop boots without a hard drive installed, you will need to replace the hard drive. The guide to do this can be found here. It isn't the best quality, but it should suffice.
If this doesn't work, try removing the WiFi and WWAN cards. Remove both cards and see if the laptop boots without them and reinstall them one at a time to see which card is causing the problem. If the WWAN card is at fault, the cards Dell used when this laptop was released are obsolete and are not worth sourcing today. Remove the card, tape the antennas to prevent shorting and fix it by removing the faulty WWAN card permanently. If the WiFi card is bad my recommendation would be to find a modern half height card you like and to purchase a half to full height bracket at the same time as your new wireless card. A word of caution is the D620/30 WiFi antennas may not reach with this adapter and a half height card, but this is easily ratified by replacing the antennas or using the antennas intended for the WWAN card. There are no guides to access these cards, but these cards are under the keyboard.
If you find none of these fixes work and your laptop is still dead (or it's another fault, like the display, inverter or video cable), I would recommend replacing the entire laptop. You don't need to buy a brand new laptop - an older refurb that's still usable today should suffice to replace your D620. What you buy is up to you, but I would recommend against buying another D620/30. I would consider a E6420 if you want something older that parts are still relatively affordable (and readily available). Dell still makes the battery for these laptops new, so finding a battery will be much easier then it is for your D620.
The primary reason I don't recommend getting another D620 is because Dell had to recall batteries in 2006 and 2009 due to faulty Sony batteries and this laptop was part of the recall. The site to check the battery was taken down, so there's no way to check through Dell's website anymore - you will likely need to err on the side of caution and buy a new battery to know with 100% certainty you are not using a recalled Sony battery from 2006-2009. This uncertainty leads to many sellers deciding to not include the battery and wash their hands of this 12 year old recall.
I am going to caution you here if you decide to try and source a battery and another D620 against my recommendation: While it appears Dell sells the battery again, it was also taken out of production at one point as well. It may not be possible to buy a OEM battery from Dell anymore. DO NOT BUY your battery from places like eBay, unless it is new. Many of these batteries are recalled (most sellers do not check), just about dead or have not been charged in years. You can be just about 100% certain none of these old OEM batteries will be safe to use, nor are they very likely to function.
I don't like making this kind of recommendation, but I firmly believe the D620/30 are on their last legs and it's usually better for owners to move onto newer hardware.