Take an inspector with you to the showing, and have them inspect the property ahead of time if possible. You will have to pay for this out of pocket (Ballpark of $500+ depending on house size, market, etc.), but it may be worth it to help your offer stand out as less risky for the seller.
This is a tricky one, because you'd have to be comfortable with getting the house even if your mortgage fell through. In general, this option is only a good idea for those who have enough cash to buy a home without a mortgage.
Depending on your unique situation, there may be other waivers you could offer the seller (appraisal, etc.) that would help your offer look stronger.
If your seller wants to get out as quickly as possible, and you are getting a mortgage, talk to your bank about how fast they could close. Putting in a 30-day or 45-day close could help sweeten the deal for the seller.
If a few offers come in around the same price, you may be able to pull on the seller's heartstrings to get them to choose you. Tell them about your background, why you love their home and want to live there, and how good of care you will take of it. These don't always work (and won't win you don't always work (and won't win you money on asking price).
Many offers are submitted with a $1,000 check, to be credited toward the traditional 20% down payment. However, you can write a bigger check, say $5,000. This check will simply be torn up if you don't get the offer, but it can show the seller you have skin in the game and really want the home.
Keep in mind, it's not over yet! Once your offer has been accepted, it's time to schedule an inspection (if you have not already had a private one done and/or waived your inspection contingency.) An inspector will accompany you to the home and look it over, top to bottom. This can take several hours.
If there are issues, you may have room to negotiate with the seller (for example, negotiate to decrease the purchase price or ask them to fix certain things before you move in). If there are too many issues, you can back out and keep hunting.
After the inspection, you will need to have an appraisal done. This is where an independent appraiser comes in and assesses what the home is worth. Many banks will not back a loan if the assessed value is significantly lower than the purchase price (though this depends quite a bit on the market you are in.)
After that, the fire department will inspect the property to ensure it is safe. They'll check that the smoke alarms and carbon monoxide alarms work, there are enough fire exits, and there are no other major safety concerns.
A typical close date is about 60 days after the purchase and sale the agreement that you will purchase the home has been signed, though either side may negotiate this. There are limits on how fast a bank can close, though, if you are getting a mortgage.
This is a time of much paperwork, so prepare yourself. Here are some examples of things your bank will ask for before the mortgage is finalized (it's a good idea to get them handy before you get serious about your real estate search):
On the close date, you will meet with your lawyer and sign a very large stack of papers. After it's all over, the home is yours!
Watch a demo of how it works!