There are several sources.
Free amino acids are present in the cytoplasm either through transport systems that can deliver enough amino acids to support growth, some are synthesized by enzymes in the cytoplasm and some are sourced out from protein degradation.
When a protein has outlived its usefulness or become damaged, it is degraded by the cell. In eukaryotes, a protein that is to be degraded has a number of copies of the small protein ubiquitin attached to it by a series of ubiquitin-adding enzymes.
Ubiquitin serves as a tag that marks the protein for degradation. A tagged protein is then sucked into a large cellular machine called the proteasome, which itself is made up of a number of protein components and looks something like a trash can.
Inside the proteasome, the tagged protein is digested into small peptide fragments that are released into the cytoplasm where they can be further digested into free amino acids by other proteases.
For further reading, go to BiologyReference.com.
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