Ok, I THINK I understand what you want now, and let me clarify to confirm before the query. You want 1 record for each user. For each user, you want their BEST POINTS score record. Of the best points per user, you want the one with the best average time. Once you have all users "best" values, you want the final results sorted with best points first... Almost like ranking of a competition.
So now the query. If the above statement is accurate, you need to start with getting the best point/average time per person and assigning a "Rank" to that entry. This is easily done using MySQL @ variables. Then, just include a HAVING clause to only keep those records ranked 1 for each person. Finally apply the order by of best points and shortest average time.
select
U.UserName,
PreSortedPerUser.Point,
PreSortedPerUser.Avg_Time,
@UserRank := if( @lastUserID = PreSortedPerUser.User_ID, @UserRank +1, 1 ) FinalRank,
@lastUserID := PreSortedPerUser.User_ID
from
( select
S.user_id,
S.point,
S.avg_time
from
Scores S
order by
S.user_id,
S.point DESC,
S.Avg_Time ) PreSortedPerUser
JOIN Users U
on PreSortedPerUser.user_ID = U.ID,
( select @lastUserID := 0,
@UserRank := 0 ) sqlvars
having
FinalRank = 1
order by
Point Desc,
Avg_Time
Results as handled by SQLFiddle
Note, due to the inline @variables needed to get the answer, there are the two extra columns at the end of each row. These are just "left-over" and can be ignored in any actual output presentation you are trying to do... OR, you can wrap the entire thing above one more level to just get the few columns you want like
select
PQ.UserName,
PQ.Point,
PQ.Avg_Time
from
( entire query above pasted here ) as PQ