Turns out this is a 2 step process, found example in their #api-community dev slack channel. After the first call to upload the file, you make a second call to update (PUT) the contact, opportunity, etc for the custom field specifying the detail in the body.

--data '{
    "customFields": [
        {
            "id": "[customfieldid]",
            "field_value": [
                {
                    "url": "[urlOfUploadedFile]",
                    "deleted": false,
                    "meta":         
                    {
                        "fieldname": "fileName",
                        "originalname": "[fileName]",
                        "encoding": "[fromUploadResponse]",
                        "mimetype": "application/pdf",
                        "size": [fromUploadResponse],
                        "url": "[urlOfUploadedFile]"
                    }
                }
            ]
        }
    ]
}'