This probably isn't the most clever way, but it'll do the job: set a counter to the number of lines you want to print out after data matches the pattern, and then 1) print out the line if the counter is positive, and 2) decrement the counter for each line that's printed out.

use warnings;
use strict;

my $nLines = 0;
while (<DATA>) {
    $nLines = 5 if /kjkjkjkj/;
    print if $nLines-- > 0;
}
__DATA__
1asdfasdf
2asdfasdf
3kjkjkjkjkj
4asdfasdf
5asdfasdf
6asdfasdf
7asdfasdf
8asdfasdf
9asdfasdf
10asdfasdf
[download]

Outputs:

3kjkjkjkjkj
4asdfasdf
5asdfasdf
6asdfasdf
7asdfasdf