How does Cro protein expressed by lambda phage kill its host?
Have a look at the wikipedia page on lambda-phage (it is nicely illustrated and referenced). 

See this map of the lambda-phage genome (Source).




Cro (Controller of Repressor's Operator) does not kill the host. It is a transcription factor that represses cI. cI, in turn, represses Cro and other pro-lytic genes (expressed from the PL promoter), thereby forming a positive feedback loop with Cro (some people also use the term "double-negative" feedback loop which in my opinion is incorrect).
During infection the virus just injects its linear DNA into the host via a porin protein (a sort of membrane channel). But at the peak of the lytic cycle many virus particles are formed inside the host cell (not just DNA but fully packed viruses). The viral proteins R and S cause breakdown of the cell membrane and the cell wall, which facilitates cell lysis.
As mentioned in the previous point, the virus is fully packaged while inside the host. This happens only during the lytic phase. During lysogeny, the viral DNA gets integrated with the host genome and remains dormant until triggered by some kind of stress such as UV (which actually leads to activation of SOS response protein RecA which by degrading cI tilts the balance towards the lytic cycle).