Take your C5 for a spin before changing the oil.  Hot oil is easier to drain.

First, start by acquiring some GM accepted oil for your C5 Corvette and an ACDelco PF44 (or equivalent) oil filter.  I chose Mobil 1.  You will need 6.5 quarts.  You will also need to jack the car up and support it safely.  Check out my HOW TO lift your C5.  You'll want to support the C5 with the rear slightly higher than the front, to allow the oil to drain quickly.  This is because the oil pan bolt is at the front of the pan.

Prepare your tools.  You will need an oil tub capable of collecting 6.5 quarts of oil, a wrench, an oil wrench, a funnel and some shop towels or rags.

Lift the hood and remove the filler cap.  This will allow the oil to flow out of the engine faster.  Jack and support your C5 using my instructions. 

Then remove the oil pan plug.  Be careful not to strip the bolt.  Many shops over-tighten this bolt.  If this is the case, use a 6 point socket and a ratchet, which is the suggested tool for this job.

Drain the oil into the tub.

While the oils is draining, check your suspension parts for problems.

Examine the rack and pinion dust covers.

And check your front spoiler.  Mine had some scraping damage, so I removed the burrs from the bottom of it with a razor blade.  You may have to bend it into position if you hit something with the spoiler.  The plastic piece is spring loaded, attached to a metal piece.  The metal piece sometimes needs adjustment to keep the spoiler looking neat under the nose of the Corvette.

After the oil has drained from the pan, locate and loosen the oil filter.

Oil will drain around the sides of the filter.  Then, remove the filter.

Be sure to wipe up any mess left behind.  Also, wipe and check the area where the filter seals to the engine.  Make sure the seal did not come loose from the old filter and attach itself to the engine block.

Fill a new oil filter up with oil, and lubricate the rubber seal.

Then attach the oil filter.  Once the filter comes in contact with the engine, give it another 3/4 of a turn.  Don't over-tighten.

Then reattach the oil pan bolt.  Do not over-tighten this bolt as you may strip the aluminum threads in the pan.  Keep in mind, there is no oil pressure at this part of the engine, so tighten this bolt just enough to make it seal.  Again, a socket is the preferred tool to use.

Next, check and wipe up any mess you made.  This is important, so that when you start you Corvette, you can check for leaks.

Then, add 6.5 quarts of oil.  Clean and install the oil filler cap when done.

Some folks recommend that you should wait about 10 minutes for the oil to fill up the oil pan. Then, monitor your oil pressure as you start your Corvette.

And, don't forget to reset your oil life monitor.  I also reset the "TRIP B" so that I know how many miles are on that oil, at any instance.