Several problems arise from the code. The first - scalability - reading a whole file into an array is kinda asking for trouble, you might run out of memory, and it'll probably take a longer while.
Another is that you're opening the temporary file for appending. Assuming the file is new, this shouldn't matter to you, but if a file by that name exists, you just added the new text to it's end, instead of rewriting from scratch. See perlopentut for a tutorial on openning files.
Yet another, which directly confronts your problem, is that the readline operator does not remove the input record seperator ($/, which is usually "\n") from read lines.
reading a file which contains "foo\nbar\n" into an array will yield ("foo\n","bar\n").
When you join $_ with your string, it'll keep the line break you put in, making $_ contain two lines.
To get around that line break you need to use chop, which takes off the last character of a string, or chomp which only removes an occurance of $/ from a string.
I think the wisest approach would be something like this:

open (FILE,"external_file.txt");
open (TEMP,">external_file.txt.tmp");

my $i = 0;

while (<FILE>){ # magical - puts every line in $_, until the end of fi
+le is reached
    if ($i == 3){ # if $i is the 3rd line
        chomp $_; # remove the line break at the end of $_
        $_ .= "my new line goes here\n"; # add the string
    }
    print TEMP $_; # print the line you just read to the temporary fil
+e
} continue { # performed at the end of every loop iteration
    $i++;
}

close FILE;
close TEMP;

unlink ("external_file.txt"); # incase rename does not clobber, delete
+ the original
rename ("external_file.txt.tmp","external_file.txt");
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-nuffin zz zZ Z Z #!perl