The process you are describing is characteristic of the lytic cycle of viral replication. In the lytic cycle, a virus infects a host cell, takes over the cell's machinery to replicate its own viral components, assembles new virus particles, and eventually causes the destruction (lysis) of the host cell to release the newly formed viruses.
In contrast, the lysogenic cycle involves the integration of the viral genetic material into the host cell's genome, allowing the virus to be replicated along with the host's DNA without immediately destroying the cell. The viral DNA can remain inactive for some time before re-entering the lytic cycle, at which point the lytic process would begin.