How do caspase proteins kill a cell?
Caspase do not directly kill the cell, but rather activate a process known as apoptosis, or programmed cell death. The programmed part is there to distinguish it from other types of cell death, such as necrosis, which are more aspecific death processes.

Coming back to caspases, they are a series of proteasis, that can activate in cascade in response to a pro-apoptotic signal. 

To simplify matters a lot, the pro-apoptotic signal first activates initiator caspases, such as caspase 8, 9, 10 (plus a series of other proteins).

These, in turn, cleave other effector caspases, which are the ones who effectively do the dirty job: caspase 3, 6, and 7.

Now there are many things happening at the same time (see this PDF from AbCam to get an idea of the complexity of the system).

At the nuclear level you have, for instance:


activation of DNA cleaving enzymes, such as CAD (caspase-activable DNAse), which is normally inactive, due to binding to its inhibitor I-CAD. The latter is a target of effector caspases and its degradation causes CAD to go in the nucleus and starting fragmenting DNA.
cleavage of PARP (Poly(ADP-ribose) polymerase), a nuclear enzyme involved in finding and repairing single-strand breaks on DNA.
cleavage of lamins (by effector caspases + other proteases), proteins that are important for the formation of nuclear lamina, and stability of the cell nucleus
cleavage of U1, a nuclear protein necessary for processing of mRNA.


Very complex events also happen at the mitochondria level: here you have a series of protein, which belong to the Bcl-2 family that control apoptosis mostly by regulating the levels of Ca2+ in the cell. There are 25 members of this family, that can be divided into two "sides": pro-apoptotic proteins such as Bax, Bak, and BAD and anti-apoptotic proteins such as Bcl-2, Bcl-xL, and Bcl-w.

The functioning of the Blc-2 proteins is very complex and I am not up-to-date with the last literature but to simplify, essentially what happens is that when the balance between anti-apoptotic and pro-apoptotic proteins is shifted towards the pro-, there is:


liberation of cytochrome c from the mytochondria into the cytoplasm, which can activate effector caspases.
induction of MPTPs (Mitochondrial Permeability Transition Pores), proteins which increases mitochondrial permeability which causes all sorts of trouble, eventually leading to swelling of mitochondria and loss of their function


Different events also bring to an increase in cytoplasmic calcium concentration, which can, for instance, induce the activity of calcium-activated endonuclease or other pro-apoptotic calcium-dependent proteins.

This is far from being an exaustive list of what happens, but it gives you the idea of the complexity of the system.