client_death is triggered with Damage event, as post, and if player is not more alive.
So, to trigger this forward, you need game to send a Damage message and that player is dead at this time.
Game triggers such message when whetever pev->dmg_take > 0 or pev->dmg_save > 0 or m_bitsHUDDamage != m_bitsDamageType. This is checked at each frame.
Calling TakeDamage should work, if damage is actually applied; as it sets pev->dmg_take at some point.
Calling Killed won't work as it doesn't set anything.