To begin, you'll need an API key from CoinMarketCap. Visit the CoinMarketCap Developer Portal, sign up for an account, and obtain your API key. This key will allow you to make requests to the CoinMarketCap API to retrieve cryptocurrency data.

Make sure you have Python installed on your computer, as well as the `requests` library, which will be used to make HTTP requests to the API. You can install `requests` by running `pip install requests` in your command line or terminal.

Use the `requests` library to send a GET request to the CoinMarketCap API. You'll need to include your API key in the request headers for authentication. Here is a sample code snippet:
```python
import requests
url = 'https://pro-api.coinmarketcap.com/v1/cryptocurrency/listings/latest'
headers = {
'Accepts': 'application/json',
'X-CMC_PRO_API_KEY': 'your_api_key_here',
}
response = requests.get(url, headers=headers)
data = response.json()
```
Replace `'your_api_key_here'` with your actual API key.

After receiving the response from the API, you should check the status code to ensure the request was successful. If the status code is 200, it means the request was successful, and you can proceed to parse the JSON response. Handle any errors or exceptions as necessary.

Extract the relevant data from the JSON response. CoinMarketCap's API returns a lot of information, so decide which data points you need. For example, you might want to extract the names, symbols, and prices of cryptocurrencies.
```python
if response.status_code == 200:
cryptocurrencies = data['data']
selected_data = [{'name': crypto['name'], 'symbol': crypto['symbol'], 'price': crypto['quote']['USD']['price']} for crypto in cryptocurrencies]
else:
print("Failed to retrieve data:", response.status_code)
```

Use Python’s built-in `json` library to convert the parsed data into JSON format. This step involves serializing your Python object (e.g., a list of dictionaries) into a JSON-formatted string.
```python
import json
json_data = json.dumps(selected_data, indent=4)
```

Write the JSON-formatted string to a local file on your system. Choose a directory and filename for your JSON file, and use Python’s file handling capabilities to write the data.
```python
with open('cryptocurrency_data.json', 'w') as json_file:
json_file.write(json_data)
```
This will save the data to a file called `cryptocurrency_data.json` in the current working directory.
By following these steps, you can successfully move data from CoinMarketCap to a local JSON file without using third-party connectors or integrations.