Warning: mysqli_connect(): (HY000/1045): Access denied for user 'username'@'localhost' (using password: YES)

That combination of username, host, and password is not allowed to connect to the server. Verify the permission tables (reloading grants if required) on the server and that you're connecting to the correct server.

You probably have an anonymous user ''@'localhost' or ''@'127.0.0.1'.

As per the manual:

When multiple matches are possible, the server must determine which of them to use. It resolves this issue as follows: (...)

• When a client attempts to connect, the server looks through the rows [of table mysql.user] in sorted order.
• The server uses the first row that matches the client host name and user name.

(...) The server uses sorting rules that order rows with the most-specific Host values first. Literal host names [such as 'localhost'] and IP addresses are the most specific.

Hence, such an anonymous user would "mask" any other user like '[any_username]'@'%' when connecting from localhost.

'bill'@'localhost' does match 'bill'@'%', but would match (e.g.) ''@'localhost' beforehands.

The recommended solution is to drop this anonymous user (this is usually a good thing to do anyways).

Below edits are mostly irrelevant to the main question. These are only meant to answer some questions raised in other comments within this thread.

Edit 1

Authenticating as 'bill'@'%' through a socket.

    root@myhost:/home/mysql-5.5.16-linux2.6-x86_64# ./mysql -ubill -ppass --socket=/tmp/mysql-5.5.sock
    Welcome to the MySQL monitor (...)

    mysql> SELECT user, host FROM mysql.user;
    +------+-----------+
    | user | host      |
    +------+-----------+
    | bill | %         |
    | root | 127.0.0.1 |
    | root | ::1       |
    | root | localhost |
    +------+-----------+
    4 rows in set (0.00 sec)

    mysql> SELECT USER(), CURRENT_USER();
    +----------------+----------------+
    | USER()         | CURRENT_USER() |
    +----------------+----------------+
    | bill@localhost | bill@%         |
    +----------------+----------------+
    1 row in set (0.02 sec)

    mysql> SHOW VARIABLES LIKE 'skip_networking';
    +-----------------+-------+
    | Variable_name   | Value |
    +-----------------+-------+
    | skip_networking | ON    |
    +-----------------+-------+
    1 row in set (0.00 sec)

Edit 2

Exact same setup, except I re-activated networking, and I now create an anonymous user ''@'localhost'.

    root@myhost:/home/mysql-5.5.16-linux2.6-x86_64# ./mysql
    Welcome to the MySQL monitor (...)

    mysql> CREATE USER ''@'localhost' IDENTIFIED BY 'anotherpass';
    Query OK, 0 rows affected (0.00 sec)

    mysql> Bye

    root@myhost:/home/mysql-5.5.16-linux2.6-x86_64# ./mysql -ubill -ppass 
        --socket=/tmp/mysql-5.5.sock
    ERROR 1045 (28000): Access denied for user 'bill'@'localhost' (using password: YES)
    root@myhost:/home/mysql-5.5.16-linux2.6-x86_64# ./mysql -ubill -ppass 
        -h127.0.0.1 --protocol=TCP
    ERROR 1045 (28000): Access denied for user 'bill'@'localhost' (using password: YES)
    root@myhost:/home/mysql-5.5.16-linux2.6-x86_64# ./mysql -ubill -ppass 
        -hlocalhost --protocol=TCP
    ERROR 1045 (28000): Access denied for user 'bill'@'localhost' (using password: YES)

Edit 3

Same situation as in edit 2, now providing the anonymous user's password.

    root@myhost:/home/mysql-5.5.16-linux2.6-x86_64# ./mysql -ubill -panotherpass -hlocalhost
    Welcome to the MySQL monitor (...)

    mysql> SELECT USER(), CURRENT_USER();
    +----------------+----------------+
    | USER()         | CURRENT_USER() |
    +----------------+----------------+
    | bill@localhost | @localhost     |
    +----------------+----------------+
    1 row in set (0.01 sec)

Conclusion 1, from edit 1: One can authenticate as 'bill'@'%'through a socket.

Conclusion 2, from edit 2: Whether one connects through TCP or through a socket has no impact on the authentication process (except one cannot connect as anyone else but 'something'@'localhost' through a socket, obviously).

Conclusion 3, from edit 3: Although I specified -ubill, I have been granted access as an anonymous user. This is because of the "sorting rules" advised above. Notice that in most default installations, a no-password, anonymous user exists (and should be secured/removed).

First of, I can't imagine the reason why you've deleted root user. But back to the question - you should specify WITH GRANT OPTION, like this:

(However this should be opted after you get the mysql re-installed as if you don't have the root access and the user is not having the sufficient privileges also, then the best is to restart the install process and make the user and grant them privileges the way defined below)

mysql> create user 'golden'@'localhost' identified by 'password';
Query OK, 0 rows affected (0.00 sec)

mysql> grant all privileges on *.* to golden@localhost with grant option;
Query OK, 0 rows affected (0.00 sec)

mysql> flush privileges;
Query OK, 0 rows affected (0.00 sec)

and then

mysql> select current_user();
+------------------+
| current_user()   |
+------------------+
| golden@localhost |
+------------------+
1 row in set (0.00 sec)

mysql> create database testing;
Query OK, 1 row affected (0.03 sec)

mysql> grant all privileges on testing.* to test;
Query OK, 0 rows affected (0.02 sec)

mysql> drop database testing;
Query OK, 0 rows affected (0.12 sec)

-but yet again, think twice before deleting root user.