The main problem is that you're trying to run a bash command from within mysql. If you run that command at the regular terminal prompt the format of the command is correct.
The second issue is that you have a zip file and not a SQL file so you need to unzip it first.
How do I load a sql.gz file to my database? (on Server Fault) explains most of what you need. My answer there should work here too with a slight modification:
unzip -p /var/www/backup.zip | mysql -u root -p mydb
unzip -p unzips to a pipe, so that you can pipe that into your database command. Make sure the zip file only contains one sql file.
mysql -u root -p mydb is going to prompt you for a password (the
-p without a value) and then attempt to pipe in the file data to
mydb - change the
mydb to your own database. You shouldn't specify the password in the command line as it then remains in your command history.