I have tried below code snippet on my local database that seems easy. Let's have a look.
Nuget Extention:
Download following reference from Nuget Package Manager On your project Dependencies part
using System.Data.SqlClient;
local.settings.json:
{
  "IsEncrypted": false,
  "Values": {
    "AzureWebJobsStorage": "UseDevelopmentStorage=true",
    "FUNCTIONS_WORKER_RUNTIME": "dotnet",
    "sqldb_connection": "Data Source=.;Initial Catalog=DatabaseName;Connection Timeout=30;Integrated Security=True;"
  }
}
Read Connection On Function Body:
//Read database Connection
        var sqlConnection = Environment.GetEnvironmentVariable("sqldb_connection");
Function Read Write Operation Using Connection String:
// Convert all request perameter into Json object
                var content = req.Content;
                string jsonContent = content.ReadAsStringAsync().Result;
                dynamic requestPram = JsonConvert.DeserializeObject<AzureSqlTableClass>(jsonContent);
                // Validate required param
                if (string.IsNullOrEmpty(requestPram.FirstName))
                {
                    return req.CreateResponse(HttpStatusCode.OK, "Please enter First Name!");
                }
                if (string.IsNullOrEmpty(requestPram.LastName))
                {
                    return req.CreateResponse(HttpStatusCode.OK, "Please enter Last Name!");
                }
                //Read database Connection
                var sqlConnection = Environment.GetEnvironmentVariable("sqldb_connection");
                var responseResults = 0;
                //Read Write Uisng Connection String
                using (SqlConnection conn = new SqlConnection(sqlConnection))
                {
                    conn.Open();
                    var text = "INSERT INTO AzureSqlTable VALUES ('" + requestPram.FirstName + "', '" + requestPram.LastName + "', '" + requestPram.Email + "') ";
                    using (SqlCommand cmd = new SqlCommand(text, conn))
                    {
                        responseResults = await cmd.ExecuteNonQueryAsync();
                    }
                    conn.Close();
                }
                return req.CreateResponse(HttpStatusCode.OK, responseResults);
Note: While publish your function on azure portal just replace the connection string on local.settings.json file. It will work
  accordingly. See the screen shot below:
