The short answer is that they are broken up in acidified bubbles inside of cells called lysosomes. Lysosomes are meant to break down either endocytosed thing (i.e. stuff that the cell "ate" by engulfing it) or things from inside the cells (a process called autophagy).

The longer answer is that there would be 2 major pathways for this:

1. If the body makes antibodies to a virus, those are proteins which stick to the virus and signal for other cells to clear out that virus. Macrophages, white blood cells which "eat" infectious material, would then endocytose or "eat' the coated virus. Inside the cells, the virus would be in a bubble called an endosome, which will eventually merge with a lysosome and digest the virus inside.

2. An infected cell can be killed by white blood cells of two main types, natural killer (NK) cells and CD8 T-Cells. NK cells recognize infected cells more generally, rather than specific infections, whereas CD8 T-Cells are specific to particular infections, and recognize bits of protein from those infections on the cell surface. When either of those types of cell recognizes an infected cell, they signal that cell to kill itself through a process of apoptosis, in which it digests its own proteins, DNA, etc... and breaks down into multiple small vesicles (bubbles), which can then be cleaned up by macrophages.

I'm sure there's exceptions to what I've said, but these would be the two main ways.

Source: I have a PhD in cell and molecular biology specializing in viruses and I used to work on autophagy in particular, so learned a lot about the terminal digestion pathways and merging with lysosomes.