How To Build A Teardrop Trailer/Trailer
Our Simple Teardrop 
For the simple teardrop we will discuss the trailer from Harbor Freight as it is the cheapest on the market and even cheaper with the store's periodic sales. The Harbor Freight item number is 90154-7VGA.
The Harbor Freight trailer in particular is a 1175 pound capacity "folding" trailer. It generally runs for about $250 on sale. It has 12 inch 4 lug spoked wheels and leaf spring suspension. The bed is 4 foot by 8 foot which is what the dimensions of the teardrop trailer will be. The trailer comes in a box and you have to assemble it - which is a good thing since it will not be assembled exactly according to its plans. The teardrop needs a non-folding trailer with the axle 40% of the distance from the rear of the trailer bed to the front of the trailer bed. The reason for making it non-folding is because the trailer will become a solid unit with the teardrop body. The reason for moving the axle is 2 fold. First, there needs to be clearance for the side doors and second, the weight of a full galley and storage area in the back will be over the axle. 40% is not absolute. You may want to change this based upon how much weight you plan to put on the front and rear of the trailer (via what you are going to build into the trailer).
To make the Harbor Freight folding 4x8 trailer a non-folding trailer, simply take the 2 crossmembers originally used in the middle (where the trailer folds) and place them inside the side members (which will be end-butted) to join the side members together. All the steel frame parts are "C" channels and the cross members will slide right into the inside of the side member as they are dimensionally smaller. Of course, there will now be an area in the middle of the frame will no cross members. Some people make a wooden cross member for floor support but its really not necessary.
The other deviation from the normal assembly is moving the axle back from the center of the trailer bed to around 40% from the rear. The good news is that the HF trailer has its leaf spring mounts on one piece of angle iron. So you can just slide the axle anywhere you want. The bad news is that you will still have to drill new holes for mounting it. You may be able to find one or two holes that line up with other holes in the frame rails to reduce the number of holes you have to drill. I can tell you that having the axle 36 inches from the rear of the trailer will give you about 120 pound tongue weight fully loaded with camping gear stored behind the axle in the galley. I would recommend mounting the axle a little farther forward than 36", however, if you want a tongue box and are going to have significant weighted gear laying on the bed while traveling.
Now would also be a good time to take a hacksaw and cut off the dolly caster mounts. They are the 6 inch pieces of angle iron perpendicularly welded to the leaf spring mounts. This is optional.
The other issue to consider is that the 2 crossmembers you used to join the side rails are now probably over-lapping the side rails where you want to drill so you'll have 2 layers of steel to drill through. You may consider cutting one of the cross members in half and use the half pieces for joining the side rails. This may avoid the overlap.
In any case no one can really tell you how and where to drill the holes as it will be unique for your trailer.
The rest of the trailer frame assembles as per the directions. You don't need to install the fenders, fender mounts, tail lights or wiring yet.
One look ahead issue you should decide at this point is whether you want frame bolts sticking through the top of the frame. The next step is making the wooden trailer bed. If you have bolts sticking up on top of the rails, you will need to make reliefs in your wooden trailer bed. The other option is to put all the frame bolts through the sides and bottom of the rails so that you have no interference on the top of the rails.
There have been cases where the trailer hubs have been shipped with little bearing grease and/or metal filings in the grease. Now would be a good time to remove the hubs, clean and repack the bearings for your own piece of mind.
Other Trailer Options 
Lowering the trailer 
Some builders of teardrops desire a more lowered look reminiscent of trailers build in the 40's and 50's. In order to achieve this they will move the axle from its attachment points on the bottom of the leaf springs to on top of the leaf springs. The axle is held on to the leaf springs with U bolts.
Axles may have a preset camber to them so it is important to keep the axle in the same orientation when it is re-mounted. In other words, the top of the axle should still be the top.
This is also a safety issue as you are stressing the U bolts in a manner that was not intended. Suspension parts have been selected by the manufacturer based on loads and strengths for the given application. Changing the application could cause a the part to fail (the U bolts in this case).
Classic custom teardrop trailer 
If you really want the classic teardrop trailer look then you really need to make or buy a trailer frame and use torsion axles. Torsion axles are simple to install with only four bolts, and are available in stock sizes from outlets such as Northen Tool and Tractor Supply, however these axles lightest load is 2000 pounds. If you are building your own Teardrop, you will probably end up wanting a custom size which is available for the same price, around $200. You can designate the width of you own trailer frame, the hub face to hub face distance you desire, and set the start angle for the trailing arm to your specification for a lower or higher stance, and de-rate the axle load to as you desire.
One option you should never consider is to build a "frameless" trailer (i.e. wooden frame). This is a dangerous practice luckily only attempted by a few (usually to be pulled only by antique cars that don't travel at highway speeds). You are making a home-made trailer for the public roadways. You don't want any chance that your trailer will fall apart on the road and injure another person.