Youre best off trying to get alot of characters onto the leaderboards whenever Asylum is the weekly trial, I doubt you need to do more than a decent vitality run without any sidebosses so you get around like 35-40k, that should be enough for it to stick onto the leaderboards.
The competition for weeklies shouldnt be that high anymore, especially not for asylum weekly, alot of people dislike that place by now.
As already mentioned, try your hardest that week to get as many characters as possible onto the leaderboards to increase the chances of getting your desired weapon, because for each character on the leaderboard you get a guaranteed perfected weapon.
The classic way would be progressing and farming asylum hardmode on veteran, but if you cant join progression guilds / groups for it then weeklies are the only other option.