Meaning of "that friendship will not continue to the end which is begun for an end"

Solution 1:

In this case

... for an end

is used to mean "for a purpose". Another example of this use case is "to what end", or "means to an end".

Knowing this, the meaning of the quote is clear:

If you become friends with someone in order to get something (or any purpose other than friendship), that friendship will not last life-long (or as the quote says, "continue to the end").

Solution 2:

That friendship will not continue to the end which is begun for an end.

It might be clearer to you if we rearrange this sentence into a more normal order:

That friendship which is begun for an end, will not continue to the end.

As others have said, "for an end" means "for a purpose (other than friendship)".

Solution 3:

Quote is easier to follow when broken in parts:

That friendship will not continue to the end

Meaning - The friendship will not last.

which is begun for an end.

Meaning - If the friendship was started for other reasons/ulterior motives..

If a friendship that started out for other(not sincere) reasons then that friendship will not last till the end.

Solution 4:

A friendship for something(money, power, etc.) will not last long.