Running Cron Job w/ Sudo permissions
If you want to specify the the root user's crontab, you have to use the -u option.

sudo crontab -e -u root

That said, there's a safer way to clean out your trash than just removing the whole directory.  Personally, I like to remove files that are older than a certain number of days.  This way, if I inadvertently delete something today, I have a "buffer" of time in which to recover from.

The following command will delete all files that are older than 15 days. 

find ~/.Trash/ -type f -mtime +15 -exec rm {} \;

Now...that said...

cron has been deprecated for launchd.  I would take some time and learn how to create .plist XML files for lauchd and use that service instead of cron