You mention wanting to print the lines above and below a matching line, but your code prints the matching line, too. In case you wanted to print all (two or) three lines, consider the following that you can adapt for files:

use strict;
use warnings;

my ( $prevLine, $nextLine );

for ( ; ; ) {
    last if eof DATA;
    chomp( my $currLine = defined $nextLine ? $nextLine : <DATA> );

    if ( $currLine =~ /match this/ ) {
        print '-' x 25, "\n";
        chomp( $nextLine = <DATA> ) if !eof DATA;

        print $prevLine, "\n" if defined $prevLine;
        print $currLine, "\n";
        print $nextLine, "\n" if defined $nextLine;
        print '-' x 25, "\n";
    }
    else {
        undef $nextLine;
    }

    $prevLine = $currLine;
}

__DATA__
The first line match this
Not this
abcdefg
The one above
Another match this 1
Another match this 2
the one below match this 2
zxcvbnn

Another match this blank above
Second to the last line
The last line match this
[download]

Output:

-------------------------
The first line match this
Not this
-------------------------
-------------------------
The one above
Another match this 1
Another match this 2
-------------------------
-------------------------
Another match this 1
Another match this 2
the one below match this 2
-------------------------
-------------------------
Another match this 2
the one below match this 2
zxcvbnn
-------------------------
-------------------------

Another match this blank above
Second to the last line
-------------------------
-------------------------
Second to the last line
The last line match this
-------------------------
[download]

The dashes are printed to show the desired output. If the first or last line is a match, only two lines are printed. If you only want the lines above and below a matching line, delete print $currLine, "\n";.

Hope this helps!

Addition: If you want to avoid printing the same line more than once--like in the example above--and have output that more closely resembles grepping the file, you can do the following:

use strict;
use warnings;

my ( $prevLine, $nextLine, %printed );

for ( ; ; ) {
    last if eof DATA;
    chomp( my $currLine = defined $nextLine ? $nextLine : <DATA> );

    if ( $currLine =~ /match this/ ) {
        chomp( $nextLine = <DATA> ) if !eof DATA;

        print $prevLine, "\n" if defined $prevLine and !$printed{$prev
+Line}++;
        print $currLine, "\n" if !$printed{$currLine}++;
        print $nextLine, "\n" if defined $nextLine and !$printed{$next
+Line}++;
    }
    else {
        undef $nextLine;
    }

    $prevLine = $currLine;
}

__DATA__
The first line match this
Not this
abcdefg
The one above
Another match this 1
Another match this 2
the one below match this 2
zxcvbnn

Another match this blank above
Second to the last line
The last line match this
[download]

The first line match this
Not this
The one above
Another match this 1
Another match this 2
the one below match this 2
zxcvbnn

Another match this blank above
Second to the last line
The last line match this
[download]