To help you with the terms, what you're looking at from above is the top plate. Frequently there are two 2x4's for a top plate. Two boards are used for added structure at the joints and extra fire proofing. This gives you about 3" of wood to drill through.
As ChrisF mentions, you should run a stud finder up and down the wall cavity near the halfway point to see if any fire blocking has been installed. This would be a single 2x4, and will almost always be in a wall if the ceilings are over 8' tall.
You'll significantly reduce the odds of insulation in the wall if you are doing this project on interior walls. You should also be avoiding any electrical in the same wall cavity (between the same two studs). You would see an outlet, switch, or wire coming out of the top plate if this were the case.
As for the actual wiring, I've used these low voltage mounting brackets for the hole in the wall. Place the bracket on the wall where you want the opening, mark the corners and sides, and cut out the drywall with a drywall saw. Test fit the bracket and file back any spots that are blocking you until you get a snug fit. Then bend the tabs back into the wall and tighten them with a pair of pliers. On that bracket, I use a modular plate (these comes with more openings if you need them) and then an ethernet jack mounted in the plate. You can also get jacks for cable, phone, stereo, etc.
Couple problems I see. First is the textured ceiling. You'll never be able to open that up and patch it without being obvious, the texture is something that just isn't done these days. Second, the exterior wall should have insulation in it, so you'll end up fishing the line through insulation and the vapor barrier.
Instead, I'd suggest running the line around the floor if you can get it to a convenient wall anywhere in that room. Use some fasteners to keep the cable at the base of the wall to run it around the room:
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Concrete walls typically have conduits where all wiring passes (edit: as pointed out by @alx9r, this might not always be the case. Sometimes wiring is cast directly into the concrete). This allows you to add additional wires to the same conduits, but doesn't help if you need to pass a wire between two places that don't have a conduit already in the wall.
Whether or not you can use the existing conduits is both a matter of convenience (are there existing conduits from and to the place you want to run the Ethernet cabling) and of wiring code - is it allowed to run Ethernet (low voltage) cabling along side other cables - high voltage (110/220 V), phone lines, TV coaxial cables, etc.
If there are no existing conduits you can use, you'd need to consider several options (all pictures from Google Images):
Run the cable along the walls, often by the skirting boards. The cable can be attached with clips to the wall. This method is easiest, cheapest and fastest, but people often don't like cables running visibly around the house and if the cables aren't properly secured they might be a tripping hazard.
Run the cables in plastic channels that are attached to the walls. This is similar to number 1, only that the channels help hold the cable better and are probably more aesthetic.
Run new conduits in the wall. This is done by carving a channel in the wall (called "wall chasing"), placing a plastic conduit in the channel, mortaring the conduit in place and patching up the channel on top of the conduit. As can be imagined, this is hard work (especially for long distances), very dirty (especially if grinding disc cutters are used to score the channel) and must be done carefully to avoid hitting other cables and pipes already in the wall. However, this will give the best aesthetic result.
If a conduit is used, fishing the new cable through the conduit is done either using preexisting cables in the conduit or with a fish tape. The new cable is attached to the preexisting cable or the fist tape (after pushing the fish tape through the conduit to the other side) and then pulled until the new cable is fully in the conduit. From my experience, if you have cables already in the conduit, it is much easier to pull them completely out and then fish the entire bundle (old cable/cables together with new Ethernet cable) back into the conduit.
Finally, Cat6 doesn't give that much of a difference in the common house over Cat5e - the extra shielding is useful with lots of cables running close together and for long distances. For home use spans are usually much shorter than the maximum 100 meters defined for Ethernet and there are only a few cables running side by side. However, Cat6 is much stiffer than Cat5e, making it quite challenging to fish it through conduits.