to put to death, to imprison for life, to punish
殛 centers on severe punitive action: putting someone to death, imprisoning them for life, or otherwise punishing them harshly.
殛 combines 歹 (death) with 亟, which likely contributes the sound and carries a sense of extremity or urgency, together suggesting the ultimate punishment.
The death radical 歹 beside 亟, which looks like a person under extreme pressure, suggests the most severe punishments: death or life imprisonment.
For キョク, imagine a criminal receiving the ultimate punishment and letting out a choked cry: kyok! The sound kyok cues キョク while the death radical reinforces the grim scene.