**Core Concept**
Cryoglobulinemia is a medical condition characterized by the presence of abnormal proteins (cryoglobulins) in the blood that precipitate at low temperatures, leading to inflammation and damage to blood vessels. The presence of monoclonal and polyclonal cryoglobulins in this patient suggests a mixed cryoglobulinemia, which can be caused by various factors including infections, autoimmune diseases, and malignancies.

**Why the Correct Answer is Right**
Given the patient's presentation of arthritis and purpura, along with histopathological evidence of cryoglobulin deposits around the vessels, it is essential to investigate the underlying cause of mixed cryoglobulinemia. The correct answer is related to a condition that can cause mixed cryoglobulinemia. **Hepatitis C virus (HCV) infection** is a known cause of mixed cryoglobulinemia, and testing for HCV is crucial in this scenario. HCV can lead to the production of cryoglobulins through immune complex formation and subsequent deposition in the blood vessels, resulting in inflammation and damage.

**Why Each Wrong Option is Incorrect**
**Option A:** Testing for rheumatoid factor would not be the primary step in diagnosing mixed cryoglobulinemia, although it may be elevated in some cases.
**Option B:** While autoimmune diseases like systemic lupus erythematosus (SLE) can cause cryoglobulinemia, testing for SLE would not be the initial step in this case.
**Option C:** Testing for cryoglobulins themselves would not be necessary, as the presence of monoclonal and polyclonal cryoglobulins is already established through laboratory examination.

**Clinical Pearl / High-Yield Fact**
It is essential to remember that HCV infection is a common cause of mixed cryoglobulinemia, and testing for HCV should be considered in patients presenting with arthritis, purpura, and cryoglobulin deposits around the vessels.

**Correct Answer:** C. Hepatitis C virus (HCV) infection