The simplest approach (conceptually, anyway) is to project the image onto a screen that is 25 to 30 feet from the pilot's viewpoint. This approach has been in use for years. Some very large systems have used a spherical screen enclosing the pilot position for 360° views. Simulators for aircraft without bubble canopies can make do with somewhat smaller screens. You only need to have screen where the pilot can see it from his position within the sim. Stewart Film sells a curved cylindrical display system of this type for virtual reality imaging, as does Motion Base. Check it out to see how it's done. It's a rather expensive, not to mention physically large, system for the recreational flight simmer, so you'll likely be looking for more affordable, more space conserving options.
A much cheaper approach (assuming you insist in keeping both eyes open) is to move the instrument images off the display that renders the external view, and to move the external view monitor farther away from you. It will still be within the range of stereopsis, but at least it will be perceptibly more distant from you than the instruments, which are clearly inside the "cabin". You can slightly improve the result by masking around the external display with a black curtain or cardboard. This way the presence of visible objects behind the monitor won't continually be reminding you that the external scene is in the room with you, rather than outside your simulated aircraft.
For a little more cost and complexity you can build a simple projection system using your existing monitor. Now before you get too excited, let me warn you that this approach produces rather mediocre results. On the plus side, it's really cheap and reasonably straightforward to try.
The basic idea is to use a simple convex lens to project an image using the light generated by your monitor. The problems are that the projected image will be pitifully dim, pretty well washed out by normal room light. The image will be upside down. Finally, (not really, but this is a big enough list of shortcomings) a simple convex lens does not focus different colors at the same focal distance. (This is called "chromatic aberration".)
You might think that your monitor would put out enough light to generate a respectably bright projected image. After all, you get a bright, easily viewed image on it even in a brightly lit room. The thing is, the monitor is designed to emit light in many different directions. Not only is it possible to clearly view the monitor straight on, you can also clearly view it more than 60 degrees off to any side. You get a view this far off front-and-center because the CRT is designed to pump light out in those directions. You get a dim projected image because only the light that happens to go directly through the lens will be focused into the projected image. Since the light emitted from the monitor's faceplate is spread all around and the lens can only intercept and focus a small portion of it, most of the light is wasted.
A partial solution to the dimness is to use a larger lens to focus more of the available light. But size comes with a price. The weight of a lens increases as the cube of its diameter. So, if you want to use a lens ten times bigger across, it will weigh a thousand times as much. Fortunately, this weight issue was solved by a guy named A. J. Fresnel with a special reduction program for overweight lenses. This program scrunches the lens down, almost flattening it out, but preserving all the really important curves, sort of like those programs aimed at women right after fattening holidays. Lenses that have successfully completed this program are called Fresnel lenses. Unfortunately, the Fresnel reducing process also slightly reduces the lens' ability to present you with a perfect image. (Optical cellulite??)
Fresnel lenses show up all over the place where large, lightweight lenses are needed and optical perfection is not. Fresnel originally developed his namesake lens for use in lighthouses where they remain on duty today. The lens used in an airport beacon is a Fresnel lens. They are used extensively in theater lighting. Plastic Fresnel lenses are used as "page magnifiers", 8 ½" by 11" rectangular magnifying lenses used to make reading a bit easier when your eyes are a bit old and tired. It's just this sort of lens that we can use to brighten up our projected image.
An 8 ½" by 11" fresnel lens has an area of roughly 90 square inches compared to about seven square inches for a three inch diameter regular convex lens. This will intercept and focus about a dozen times the amount of light, giving us a considerably brighter image, but still not bright enough to view in a bright room.
To maximize the perceived brightness of the projected image, we have to use this projection system in a dark room. Any extraneous light will reduce the contrast of the projected image and give it a washed out appearance. Extraneous light is any light that doesn't go directly from the monitor through the lens. To reduce extraneous light from the monitor itself, mount the lens in the end of a box that encloses the front of the monitor. The box can be made in two telescoping pieces to allow focusing. To further increase the image contrast, reduce light reflections inside the box by painting the inside flat black.
The size of the box depends on the focal length of the lens, and, of course, on the size of the monitor. The focal length of a lens is the distance from the lens to the image the lens forms of a light source that is a great distance away. For use in a projector the lens should be placed at least one focal length away from the monitor surface (where it will project an infinitely large, infinitely dim image at infinity), and no farther than two focal lengths away (where is will project a one-to-one image, two focal lengths away from the lens). For a simple projection system like this the distances between the lens and the monitor, and the lens and the projection screen are related by a simple equation: 1/d1 + 1/d2 = 1/Fl. "d1" refers to the distance between the monitor surface and lens. "d2" refers to the distance between the lens and the projection screen. "Fl" refers to the focal length of the lens. It doesn't matter what units are used as long as they are all the same.
One last point. The projected image will be inverted. The readily apparent, easy fix is to turn the monitor over. This works optically, but maybe not thermally. Monitors are made to be quiet and cheap. Typically they have no fan, relying instead on convection to move cooling air through them. Heat generating components are carefully located so nothing sensitive is located above them. You can probably see where I'm going. Once flipped, the monitor can now start cooking itself. Will it happen to your monitor? Maybe, maybe not. To be on the safe side, thermally anyway, consider using a small cooling fan to force air through the monitor.
Fresnel lenses are available from a number of sources. You can buy an intermediate quality Fresnel lens, called a "full page magnifier", from Office Depot. Fresnel lenses show up from time to time on Ebay. Edmund Industrial Optics and JML Optical Industries both carry a large assortment of higher quality, and of course more expensive, Fresnel lenses.
Building a projection system based on an LCD panel is not totally out of the question. Early models of portable business presentation projection systems simply used a large LCD panel as an electronic transparency for use on an overhead projector. There is no reason you couldn't do something very similar. Newer systems function more like a slide projector. Smaller units may have a single color LCD panel while larger, brighter units use multiple, monochrome panels, each illuminated by a particular color of light. Using multiple LCDs allows higher resolution panels to be used as well as limiting the amount of heat each panel picks up from the light source. Temperature management is a big deal in LCD projectors. Let the panel get too hot and you're got, well, toast! That's why the commercial systems all have fans.
If building an LCD projector appeals to you, you might start by visiting the Audio Visualisers site. There is a do it yourself projector project page that includes a list of links along with some general project info.
Stepping up to a commercially available flat screen projection system brings a significant boost in realism at a significant boost in cost. SVGA projectors are coming into reach as prices for the smaller units approach the US$1,000 mark. Don't get scammed buying a used projector that "only has a burned out projector bulb". One might think a bulb would only cost a couple of bucks (quid?, euros?). That might be true, but sadly, one cannot simply replace the bulb. The bulb is part of a precisely aligned unit that must be replaced in total at a cost measured in the hundreds of dollars.
If you would like to learn more about the design of projection systems you should do a little reading about optics and optical design. Optics books come in two varieties. There's the insidious college level variety that is full of very complex math and terms like "modulation transfer function". It's insidious not because of the math, but because after even a brief exposure, you will find yourself inventing occasions to drop terms like "modulation transfer function", even when you can no longer recall what they mean. The second variety refers to "geometrical optics". It doesn't use such heavy-duty math, relying instead on ray tracing diagrams accompanied by basic trig functions. Further you won't find yourself dropping terms like "modulation transfer function". As a starter take a look at the Edmund Industrial Optics web site, where they have posted a tutorial on geometrical optics. After that, it's time for a trip to the library. If you can't find a book specifically addressing optics, check into a first year college physics textbook. Often these books have a chapter on simple optics. There's a pretty good discussion to be found in the Feynman series of physics lectures.
Besides commanding a fairly stiff price, projection systems also require a relatively large amount of room to install. A more compact alternative is to use a collimated display system. These systems make use of optical components, basically lenses and/or mirrors, to make the perceived image appear well beyond the range of stereopsis.