no start on 2001 chevrolet suburban 1500 5.3

Does it crank but not start? If so I bet it's the FPDM. Which is your Fuel Pump Driver Module. I just replaced one on a Suburban about a week ago.

you could also do a test at the fuse box itself. I know you did a relay swap, but first make sure the relay itself is good, lots of youtube vids on how to test them. then in the fuse box where the pump relay plugs in, locate the hole where pin number 30 from the relay plugs in. it should have 12 volt power all the time, (test with a multimeter), if not then there is no power getting to the fuse box at all and you can go from there backward to locate the problem. now, if there is power at pin 30, use a jumper wire and go from 30 to hole number 87. this is the "load" or what ever the relay is operating. you have now bypassed everything, and your fuel pump should cycle. if not, then the problem lies in that end between the relay and the pump itself.

It's located above the spare tire on the drivers side. You will need to drop your spare tire if you have one an to take it off you will need a 13mm socket. The FPDM, you will assume its the fuel pump but its not. It will cause the same symptoms like its not getting gas an just crank.

How to get the suburban 2001 back door window back up

Mind if I ask how you fixed the problem?