Granting access to all tables for a user on PostgreSQL Database

First, you have to be able to connect to the database in order to run queries. This can be achieved by

REVOKE CONNECT ON DATABASE your_database FROM PUBLIC;

GRANT CONNECT
ON DATABASE database_name 
TO user_name;

The REVOKE is necessary because

The key word PUBLIC indicates that the privileges are to be granted to all roles, including those that might be created later. PUBLIC can be thought of as an implicitly defined group that always includes all roles. Any particular role will have the sum of privileges granted directly to it, privileges granted to any role it is presently a member of, and privileges granted to PUBLIC.

If you really want to restrict your user to DML statements, then you have a little more to do:

REVOKE ALL
ON ALL TABLES IN SCHEMA public 
FROM PUBLIC;

GRANT SELECT, INSERT, UPDATE, DELETE
ON ALL TABLES IN SCHEMA public 
TO user_name;

These assume that you will have only one schema (which is named 'public' by default).