To pass many rows efficiently to a stored procedure, use a table-valued parameter. The C# app can specify parameter type structured and a value of a data table. For maximum performance, make sure the DataTable column types match the server-side table type column types (including max length for string columns).
Below is an example excerpt from the documentation link above:
// Assumes connection is an open SqlConnection object.  
using (connection)  
{  
  // Create a DataTable with the modified rows.  
  DataTable addedCategories = CategoriesDataTable.GetChanges(DataRowState.Added);  
  // Configure the SqlCommand and SqlParameter.  
  SqlCommand insertCommand = new SqlCommand("usp_InsertCategories", connection);  
  insertCommand.CommandType = CommandType.StoredProcedure;  
  SqlParameter tvpParam = insertCommand.Parameters.AddWithValue("@tvpNewCategories", addedCategories);  
  tvpParam.SqlDbType = SqlDbType.Structured;  
  // Execute the command.  
  insertCommand.ExecuteNonQuery();  
}
Here are T-SQL snippets to create the table type and proc.
CREATE TYPE dbo.CategoryTableType AS TABLE  
    ( CategoryID int, CategoryName nvarchar(50) );
GO
CREATE PROC dbo.usp_InsertCategories
AS
@tvpNewCategories dbo.CategoryTableType READONLY
INSERT INTO dbo.Categories (CategoryID, CategoryName)  
    SELECT nc.CategoryID, nc.CategoryName FROM @tvpNewCategories AS nc;
GO
Even for trivial inserts, TVP performance can provide performance similar to SqlBulkCopy (many thousands per second) with the advantage of a stored procedure interface.