What tests can be performed to test the purity and quality of the raw peptide HCG (Human Chorionic Gonadotropin)

A monograph from a pharmacopeia gives a good starting point for active pharmaceutical ingredient characterization. AFAIK there is no monograph for hCG nor your actual peptide of interest, but there is one for FSH (here for the european pharmacopeia monograph), which shares a subunit with CGA, meaning that modifications of it could suffice (though validating these modification will be a good amount of work, and will depend on the ultimate use for the peptide).

First, it is necessary to stress that even if the compound passes all assays, it would still be unsuitable for human use. It may be acceptable for animal experimentation. This is because the pharmacopoeia assumes a context where you have a non-adversarial relationship with your supplier, such that statement like "up to XX% remaining impurities" is an acceptable standard as the synthesis route & typical impurities are known. However, buying peptides from a random Chinese seller with no transparency on process is likely an adversarial situation.

Some keys to read the monograph:

• Compound followed by the letter R ("water R", "glycine R"): these compounds have their own monograph. You could assay them accordingly, but for this use case understanding the R as "analytical grade" is possibly good enough.
• "Follitropin CRS": chemical reference substance, a reference standard powder of follitropin obtained through reliable means (typically through EDQM for pharmaceutical purpose). Here also, substituting for an analytical grade standard from a reputable chemical seller could work.
• Method followed with three numbers ("Bacterial endotoxins (2.6.14)"): reference a method monograph found in the given chapter/section. Apply methodology as described in the referenced text.

To adapt the assays to your peptide of interest, you need to understand which parts are specific to follitropin (and should be modified), and which applied to gonadotrophin-like peptides in general. For instance, for isoelectric focusing, the running conditions (pH, catholyte/anolyte) will be adequate, but the follitropin reference will be replaced with a CGA reference. On the contrary, the selective cleavage of peptide bonds assays is specific to the peptide, and will need extensive rework.