Don't try to make your company "price match" for you.
Here's what you need to do, step by step:
1) Begin job search, don't tell ANYONE about it, except maybe a spouse and close family (who might be able to help in the search)
2) Interview and all that good stuff, wait for an offer that is truly good.
3) a) If you are satisfied with the company you are at, speak to your boss (schedule a formal meeting if you can) and tell him you would like a raise to what the amount the other company is offerring.. DO NOT MENTION that the amount is from another company or job prospect. Make sure you tell him the 5W's and H about why you deserve the raise.
3) b) If you aren't satisfied with the company you are at and think the new job is more prosperous... put in your two weeks notice. Skip to step 5.
4) If you offer is declined, put in your two weeks notice and accept the other company's offer. If your offer is low-balled, decide whether the difference in salary is enough to make you stay and pass up a new opportunity.
5) Enjoy which job you selected.
*Key points: Do NOT mention a different job to ANYONE remotely close to the workplace. Name your price, you'll be disappointed if what you get is not what YOU want and will have to wait another year to bring it up again.
Raises are not a bidding war. Forcing people to pay you a certain amount or lose you ALWAYS ends bad, unless you are that critical to the company that they CANNOT replace you for ANY amount. Even if the offer is accepted, it will leave a sore spot on your image within the company. Oftentimes they'll just tell you to enjoy your new job and escort you to the door.