Measure the distance from your computer to your outside antenna location. Be sure to choose a route that will protect your coax run and allow easy upgrading. Use a long extension cord to simulate your coax and help make the measurement more realistic. Try to keep the coax cable run as short as possible to minimize the RF power loss, and avoid sharp bends which can damage the coax, or in some cases, change the coax's impedance. Next, choose a high quality brand of coax and cut it to that distance, plus about one meter extra, for slack. Avoid the low quality, high loss cable such as RG-58, CB or TV/satellite coax. The LMR series from Times Microwave is highly recommended. The coax you choose must have a 50 Ohm impedance. Any other impedance, such as 75 Ohms, may cause too much loss for this application. Use of the 75 Ohm hardline used in cable TV service might be possible, though some experimentation will be required. You may also be able to pick up very high quality hardline coax from cellular phone or broadcast installations. They usually give the spool end runs (the last 15 meters or so) away for free or even throw it away because it's of no use to them. The connectors for commercial hardline are very expensive and hard to make, so remember that.
Here is a little additional note on using 75 Ohm cable TV hardline in your system.
Prepare the coax for installation of the two N plugs on each end. Here is a diagram to help with the installation of the two piece style N connectors. Be aware you'll need Belden 9913 style N connectors due to the large center conductor if you use LMR-400. You may also need to route your coax prior to installing the N connectors if it has to run through small spaces.
Take your time installing the connectors. At microwave frequencies, any loss is a significant loss. You should assume about 0.25 dB loss per connector or adapter that you have in your coax run. Be sure to solder to the coax's ground shield through the solder holes in the side of the connector. Try also to get a good mechanically solid installation so any stress on the coax won't cause to connector to come loose. A picture of a two piece N plug installed on LMR-400.
After soldering on the connectors, wrap some 3M Scotch Super 33+ electrical tape, or equivalent stretchy tape, around the bottom of the connector and a few centimeters of coax. Pull the tape as tight as possible to help make the connector waterproof. If your antenna is going to be exposed directly to heavy rain or high humidity, you may want to consider wrapping the connection with some self-fusing silicone tape (Radio Shack part number 64-2336). You can also view this file on sealing & waterproofing hardline connectors.
Next, route your coax to your outside antenna. Secure the cable to keep it from flopping around in the wind or from people pulling on it. Leave a small loop of coax where it enters the building. This will act as a drip loop and will keep rain from seeping into your building. You may also at this time want to install a PolyPhaser brand of inline coax lightning protection, which is highly recommened. Refer to their documentation on the proper grounding techniques that will be required for lightning protectors to work.
Connect the cable up to your antenna's N jack. Wrap Coax Seal or any other pliable waterproof tape around the connector and then also wrap all that with Super 33+ electrical tape. Secure the wrapped connector against the antenna mast. Here is a little tip if you do use Coax Seal - Don't apply it to the connector directly!, you'll just end up with a mess. Instead, wrap the connection with electrical tape then wrap the Coax Seal around the tape, sealing any small holes. This will save you alot of hassle when you have to remove the connection
On the other end of the coax, obtain a TNC plug to N jack. You can order them from Hutton Communications, part number RFT-1234, for $13.32, or Newark, part number 92N3389, for $16.05. Use this adapter to connect your coax to your modified Symphony. You may also want to make some type of stress relief holder to support the weight of the coax and adapter.