Best way to apply a "friend system"-like table?
First approach is the SQL approach. You need to make modification though. The relation that you are describing is called many to many - each player can have each other player as friend. To achieve this you need a separate table which connects player with his friends (you just need 2 IDs):

|Player_ID  | Friend_ID|
------------------------
|  1        |  2       |
|  1        |  5       |


This is the better approach if you need make a lot of queries on Player and list of his friends (this will give speed of queries).

If you just need list of names, the other approach is also possible.

So you have to think and decide which is better approach for your needs.