From the description you provided, i suspect a hardware failure. In most cases it could be either RAM or HDD . Follow the below steps and report the result in next reply.
Warning- Touch your hand in an unpainted metal part to discharge static electricity before removing any component.
Here is the link to the service manual of your laptop - ftp://ftp.dell.com/Manuals/all-products/esuprt_laptop/esuprt_latitude_laptop/latitude-d620_service%20manual_en-us.pdf
1 ) Remove the hard disk and check if the system can enter in to the BIOS. Refer page 18 of the manual if you don't know how to remove the HDD.
2 ) Check if there is an additional RAM in the DIMM socket B which is located in the bottom part. Remove it and check if the system works. Refer page 23 if you are in doubt.
3 ) If the above steps didn't helps, you need to take out the RAM in DIMM slot A which is located under the keyboard. Refer page 22 carefully if you are in doubt. Now remove and reinsert the RAM module several times to resolve any potential contact issues.
Reply after doing the above steps.