Metaphor:

In this poem, the speaker uses an "icy hand" (line 4) being laid as a metaphor for dying. The personification of death is a common image used in poems. That somebody—in this case, a king—is chosen when they die (by a character known as Death) is reinforced when the speaker has that character's hand touch the king.

Metonymy:

The "icy hand" is an example of metonymy: the hand of Death is used to refer to the whole character. Another interesting piece of metonymy in this poem is referring to the king as "sceptre and crown" (line 5): two of the inanimate objects used by kings are referring to the whole of the person. When the speaker mentions these items "must tumble down" (line 6), they are referring to the king dying and perhaps literally falling over and the sceptre and crown tumbling to the ground. But the speaker is also alluding to the tumbling of empires: often when a king dies, entire empires can change borders or even fall. In this way, then, the sceptre and crown refer to the idea of an individual king, but the king then refers to the idea of an empire or kingdom.