Begin by setting up access to the Twilio API. Log into your Twilio account and navigate to the "API Keys & Tokens" section. Generate a new API key and note down the Account SID, Auth Token, and the API Key SID. These will be used to authenticate and make requests to Twilio's API endpoints to fetch the data you need.

Use a programming language like Python to send HTTP requests to Twilio's API. Choose the specific API endpoint based on the data you need (e.g., messages, calls). Use the `requests` library to send GET requests, including your Account SID and Auth Token for authentication. Parse the returned JSON data to extract the information you need.

Install the necessary PostgreSQL client library in your programming environment to interact with your PostgreSQL database. For Python, you can use `psycopg2` or `asyncpg`. Install it using pip:
```
pip install psycopg2
```

Set up a connection to your PostgreSQL database. Use the credentials for your database server, including the database name, user, password, and host. For example, using `psycopg2` in Python:
```python
import psycopg2
conn = psycopg2.connect(
dbname="your_dbname",
user="your_user",
password="your_password",
host="your_host"
)
```

Transform the fetched data into a format suitable for PostgreSQL insertion. This step might involve cleaning the data, converting it into tuples, and ensuring that it matches the structure of your PostgreSQL table. Ensure that the data types in your database schema align with the data being inserted.

Use SQL INSERT statements to add the data into your PostgreSQL table. Create a cursor object from the database connection and execute the insert command for each data entry. Here's an example using `psycopg2`:
```python
cursor = conn.cursor()
for data in data_entries:
cursor.execute(
"INSERT INTO your_table (column1, column2) VALUES (%s, %s)",
(data['field1'], data['field2'])
)
conn.commit()
```

After the data has been inserted, close the database connection and handle any potential errors. Implement try-except blocks to catch exceptions during API requests or database operations. Always ensure that the database connection is closed in a `finally` block:
```python
try:
# Data fetching and insertion logic
except Exception as e:
print(f"An error occurred: {e}")
finally:
cursor.close()
conn.close()
```
By following these steps, you can efficiently transfer data from Twilio to a PostgreSQL database without relying on third-party connectors or integrations.