If a person received a dishonorable discharge from the military, would there be a,record of their service?

FYI, typically getting kicked out of the Army doesn't equate to a dishonorable discharge. There are 5 discharges, 3 administrative and 2 punitive, of which a dishonorable discharge is one of them. The punitives can only be given as punishment when a soldier is found guilty at a courts-martial. "Getting kicked out" generally means receiving an administrative separation. These separations can only give an administrative discharge.

you only get a dishonorable discharge if you commit a felony

like murder, rape, etc

then you go to prison

2. military records are sealed

IE: you need the written permission of the ex service members to be able to access thier military records

All military records are at The National Personnel Records Center, Military Personnel Records (NPRC-MPR)

See the link below for information on how it get records.

If you receive a dishonorable discharge it means you have a felony conviction. Unless your relative is a convicted felon his discharge, assuming he served, was probably an administrative discharge.