I can try, but a quick disclaimer: I am so not an expert and moreover it’s been over a year since my last literature course. But let’s give it a go.
What Claudius did and the consequences of his action, as well as a handful of other abstract concepts (e.g. guilt, intent, a person’s faults) are personified, i.e. he talks about them as if they were people. They have human features (hands, a face, teeth) and they can perform actions (defeat, shove, buy):
'O, my offence is rank it smells to heaven;'
'My stronger guilt defeats my strong intent'
'Whereto serves mercy
But to confront the visage of offence?’
'In the corrupted currents of this world
Offence’s gilded hand may shove by justice’
'And oft 'tis seen the wicked prize itself
Buys out the law’
'the teeth and forehead of our faults'
Although the first one is a bit questionable, it doesn’t really come across as a personification unless you analyze it in the light of what is said later in the soliloquy (offence is given hands and a face). The reason why it isn’t so straightforward is that having a smell is hardly an exclusively human quality. I think you could argue that in the first line Shakespeare is using foul smell as a metaphor for Claudius’s foul deed. It’s not the clearest example of a metaphor (cf. my offence is a foul smell) but it is implied. I’m not sure if this kind of device has some other name (making abstract concepts concrete things that can have qualities such as smell), so in my limited knowledge I’d call it a metaphor.
'It hath the primal eldest curse upon't,
A brother’s murder.’
are most likely a literary allusion to Cain and Abel. In Christian mythology the first murder ever committed (the primal eldest curse) was Cain killing his brother, or at least I think so. I’ve never studied this in any detail so I only have a very cursory knowledge of their story. I’m just going on what I’ve picked up from popular culture and a quick glance through the Wikipedia article.
'Of those effects for which I did the murder,
My crown, mine own ambition and my queen.’
Here there are several things going on. My crown, mine own ambition and my queen are synecdoches to the power Claudius has in his position as the king, i.e. they parts referring to the whole (Well, okay, I admit I might be pushing it with mine own ambition but I’m just just gonna roll with it now.) There are also three of them which is probably deliberate, repetition in threes is a very common rhetorical device. There’s also another instance of repetition:
'Try what repentance can: what can it not?
Yet what can it when one can not repent?’
'O wretched state! O bosom black as death!
O limed soul’
There’s a quite interesting juxtaposition going on between these two lines:
'Bow, stubborn knees; and, heart with strings of steel,
Be soft as sinews of the newborn babe!’
Claudius compares his stubborn knees (personification!) and his heart with strings of steel to sinews of the newborn babe. Whereas the stubbornness of his knees (i.e. they are unwilling to kneel) and his heartstrings of steel (cold, unfeeling) symbolize his wickedness, the baby’s soft sinews stand for innocence. If Claudius’s knees and heart could be as soft he might be able to repent (‘All may be well’). What I find especially fascinating is that these lines portray the idea that wicked deeds corrupt your body - if you lead a corrupted life, it will leave marks on you. (Like, you know, in The Picture of Dorian Gray). Until fairly recently this used to be quite a prominent theory and was still thrown around in the media storm surrounding Jack the Ripper in the late 19th century (although not actually taken seriously by the investigators). Soft as sinews of the newborn babe is, by the way, a simile.
And I spy with my little eye other similes as well:
'What if this cursed hand
Were thicker than itself with brother’s blood,
Is there not rain enough in the sweet heavens
To wash it white as snow?
'O bosom black as death!'
White and pure snow is yet another symbol for innocence, whereas bosom black as death symbolizes Claudius’s corruption. Black as death fascinates me because it makes me think of the Black Death, the bubonic plague, which medieval medical writers thought was caused by breathing corrupted air, so it works as a symbol for Claudius’s corruptness on that level as well.
And finally, I might be going slightly off-topic, but here’s my favourite part, which parallels Hamlet’s To be or not to be speech:
'And, like a man to double business bound,
I stand in pause where I shall first begin,
And both neglect.’
I really enjoy the way these lines invite you to draw comparisons between Hamlet and Claudius, and how they tie in the theme of action vs. inaction, being stunned because there’s no way of knowing beforehand where your actions will lead you.
(It’s 4 am where I live. If none of this makes any sense then that’s why.)