First, you need to access the database where PostHog stores its data. PostHog typically uses PostgreSQL as its backend database. Use the appropriate credentials to log into the PostgreSQL database via a command-line interface or a database management tool like pgAdmin.

Determine the specific data you need to export. This could be event data, user data, or any other analytics stored in PostHog. You can use SQL queries to explore and select the datasets you are interested in. Ensure your queries are optimized to pull only the necessary data to avoid unnecessary load.

Write an SQL query to extract the required data. For example, if you want to extract event data, your query might look like this:
```sql
SELECT FROM events WHERE created_at >= '2023-01-01';
```
Adjust the query parameters to fit your data requirements, such as specific date ranges or event types.

Use PostgreSQL's built-in capabilities to export the result of your SQL query to a CSV file. This can be done using the `COPY` command:
```sql
COPY (SELECT FROM events WHERE created_at >= '2023-01-01') TO '/path/to/events.csv' WITH CSV HEADER;
```
Replace `/path/to/events.csv` with the desired path on your local machine.

Use a scripting language like Python to convert the CSV file into a JSON file. Here's a simple Python script to perform this conversion:
```python
import csv
import json

csv_file_path = '/path/to/events.csv'
json_file_path = '/path/to/events.json'

# Read CSV and convert to JSON
with open(csv_file_path, mode='r') as csv_file:
csv_reader = csv.DictReader(csv_file)
data = [row for row in csv_reader]

# Write JSON data to file
with open(json_file_path, mode='w') as json_file:
json.dump(data, json_file, indent=4)
```

After conversion, verify the integrity and structure of the JSON data. Open the JSON file and check that the data is correctly formatted and includes all the necessary fields. Use tools like JSON validators or formatters to ensure the JSON is well-structured.

For repeated tasks, automate the extraction and conversion process using a script or cron job. This can be done by creating a shell script that runs the SQL export and Python conversion sequentially, and then scheduling it with a cron job on Unix-based systems or Task Scheduler on Windows.

Following these steps will allow you to move data from PostHog to a JSON file effectively without the need for third-party integrations.