Begin by setting up your environment to include the necessary tools and frameworks. Ensure you have Apache Iceberg installed and configured. You also need Python installed on your system since we'll use it to interact with the Coin API and for data processing.

Register on the Coin API website to obtain an API key. This key will be required to authenticate your requests to the API. Ensure you have access to the specific data endpoints you need for your project.

Use Python's `requests` library to fetch data from the Coin API. Construct HTTP requests to the desired endpoints using your API key for authentication. For example, to get exchange rates, you might use:
```python
import requests

headers = {'X-CoinAPI-Key': 'YOUR_API_KEY'}
response = requests.get('https://rest.coinapi.io/v1/exchangerate/BTC/USD', headers=headers)
data = response.json()
```
Handle any exceptions or errors in the HTTP response to ensure robustness.

Once you have the data, you need to transform it into a format suitable for Apache Iceberg. Typically, this involves converting data into a tabular format like Pandas DataFrame:
```python
import pandas as pd

# Example transformation
df = pd.DataFrame([data]) # Assuming data is a dictionary
```

Define the schema for your Iceberg table to match the structure of your transformed data. This involves specifying column names, data types, and any partitioning strategies that may be useful for your data:
```sql
CREATE TABLE my_iceberg_table (
exchange_rate DOUBLE,
asset_id_base STRING,
asset_id_quote STRING,
time TIMESTAMP
)
```
Ensure the Iceberg table is created in your configured metastore.

Write the transformed data from the DataFrame directly to your Iceberg table. This will require using Iceberg's write capabilities, typically through a supported engine like Apache Spark.
```python
from pyspark.sql import SparkSession

spark = SparkSession.builder \
.appName("IcebergWrite") \
.config("spark.sql.catalog.my_catalog.type", "hadoop") \
.config("spark.sql.catalog.my_catalog.warehouse", "path/to/warehouse") \
.getOrCreate()

# Convert Pandas DataFrame to Spark DataFrame
spark_df = spark.createDataFrame(df)

# Write to Iceberg
spark_df.writeTo("my_catalog.my_iceberg_table").append()
```

Once the data is written to Iceberg, verify the integrity by querying the table and confirming that the data matches what you expect. Use SQL queries through Spark or any other compatible query engine:
```python
result = spark.sql("SELECT * FROM my_catalog.my_iceberg_table")
result.show()
```
Check for consistency, completeness, and correctness of data to ensure the process was successful.

By following these steps, you will have successfully moved data from Coin API to Apache Iceberg without utilizing any third-party connectors or integrations.