I think you could try a number of approaches. Two I can think of are:
1. Concatenate the two columns:
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SELECT CONCAT(first_name,' ',last_name) AS name FROM table WHERE CONCAT(first_name,' ',last_name) LIKE 'name_entered_in_text_box';
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NB - if you want exact matches don't use LIKE, just use =. If you want more results, put some wildcards (%) in your LIKE clause (ie LIKE '%henry%').
2. Search each column for the same term:
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SELECT first_name, last_name FROM table WHERE first_name LIKE '%henry%' OR last_name LIKE '%henry%';
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This will give lots of results as used % before and after and I used OR. Cut the results by using fewer '%' and using AND instead of OR.
Hope that sheds some light on your problem! Good luck!
Henry