chomp(my $filename = <STDIN>);
my $dir = getcwd;
my $backupdir = 'backup_files';
my $back = $filename . ".bak";
my $backful = $dir . "/" . $backupdir . "/" . $back;
[download]

May I suggest slightly cleaner alternatives?

use File::Basename        qw( basename );
use File::Spec::Functions qw( catfile );

chomp(my $filenameful = <STDIN>);
my $filename  = basename($filenameful);
my $backupdir = 'backup_files';
my $back      = $filename . ".bak";
my $backful   = catfile($backupdir, $back);
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use File::Basename        qw( basename );
use File::Spec::Functions qw( catfile rel2abs );

chomp(my $filenameful = <STDIN>);
$filenameful  = rel2abs($filenameful);
my $filename  = basename($filenameful);
my $backupdir = 'backup_files';
my $back      = $filename . ".bak";
my $backful   = rel2abs(catfile($backupdir, $back));
[download]

They both allow a path to be attached to the input.
They both use the right directory seperator for the platform being used.
They both cannonize the path, so foo//bar would become foo/bar.
The first creates a relative path while the second creates an absolute one.