Bulk Insert Through Network

I'm going to make a leap of faith and assume you connect to SQL Server using Windows Authentication. In such a case you have a Kerberos double hop: one hop is your client application connecting to the SQL Server, a second hop is the SQL Server connecting to the remote \\COMPUTER01. Such a double hop falls under the restrictions of Constrained Delegation and you end up accessing the share as Anonymous Login and hence the Access Denied.

To fix the problem you need to enable constrained delegation for the SQL Server service account. This blog How To: SQL Server Bulk Insert with Constrained Delegation (Access is Denied) has an example of how to do it, and I really do hope that the step on how to 'enable unconstrained delegation' is just a typo as unconstrained delegation is just plain evil.

In the case I jumped the gun and you are logging into SQL Server using SQL Authentication you will need to create a credential for your SQL login and use that to access network resources. See CREATE CREDENTIAL.