Begin by ensuring your local environment is prepared. Install Python (if not already installed) as it will be used to fetch data from CoinGecko. Also, set up Apache Iceberg on your data processing platform, such as Apache Spark or Flink, ensuring all necessary Iceberg dependencies are correctly configured.

Utilize the CoinGecko API to obtain cryptocurrency data. Use Python's `requests` library to send HTTP requests. An example request to fetch data might be:
```python
import requests

response = requests.get('https://api.coingecko.com/api/v3/coins/list')
data = response.json()
```
This retrieves a list of all available coins with their respective IDs, which you can use to fetch more detailed data.

Once you have the JSON data from CoinGecko, transform it into a tabular format suitable for Iceberg. You can use Python's `pandas` library to convert JSON data into a DataFrame:
```python
import pandas as pd

df = pd.json_normalize(data)
```
Ensure that the DataFrame's schema aligns with the schema you will define in Iceberg.

Define the schema for your Iceberg table. This involves specifying the data types and structure matching your DataFrame. You can use SQL or programmatically define it using Iceberg's API in the language supported by your platform (e.g., Scala or Java).

Use your data processing platform to write the DataFrame to Iceberg. If using Spark with Iceberg, convert the DataFrame to a Spark DataFrame and write it using:
```python
spark_df = spark.createDataFrame(df)
spark_df.write.format("iceberg").mode("overwrite").save("path/to/iceberg/table")
```
Ensure the write operation matches the schema and partitioning strategy defined earlier.

After writing the data, verify that it has been correctly stored in Iceberg. Query the Iceberg table using SQL or your data processing platform’s API to ensure data integrity and correctness:
```sql
SELECT * FROM iceberg_table_name LIMIT 10;
```
This verification step is crucial to ensure that the data migration has been successful.

With the process verified, automate the workflow using a script or a cron job. This ensures regular updates to your Iceberg tables as new data becomes available on CoinGecko. Python scripts can be scheduled using `cron` in Unix-based systems or Task Scheduler in Windows.

By following these steps, you can successfully migrate data from CoinGecko to Apache Iceberg without relying on third-party connectors or integrations.