When a person is given a lethal injection in a prison they're usually given an injection first that relaxes them and puts them to sleep temporarily.
This is so that they do not hopefully feel any pain or could try to resist and then they are injected with several different drugs which then stops the heart.
Usually if the lethal injection is done correctly the person should experience a painless death within a few minutes however there have been some botched lethal injections where it took an hour or more for the person to die and they died in a painful way.
Most cases of lethal injection the death happens within 7 minutes though and death is pronounced after cardiac activity stops.
However in rare cases the whole procedure to give a lethal injection can take up to two hours, as was the case with the execution of Christopher Newton on May 24, 2007.
If they find a suitable vein quickly then the lethal injection and death usually happens pretty quickly.