Plug the standard phone into an ethernet jack

First of all, you can only have one modem/router on a single phone line. So if your jack in the bedroom is hooked up to the same phone line as the one in the living room then you can't get a new modem/router to have in your room.

You say there are two jacks in the living room. If one of them is used for hooking up the modem/router then what is the other one used for? Could that perhaps be connected to the jack in your bedroom? You can find that out by getting a phone line tracer (device that sends out sine waves over copper wires and its corresponding receiver) hook that up to your bedroom jack and take the receiver to the living room and see if it sees the signal. If you don't hear the signal there, then that means that the copper line connected into your bedroom has no connection with the copper lines connected in your living room and there is no way to get any signal between the two places using the preinstalled copper lines.

If you do hear the signal in the living room that means the jack in your bedroom is hooked up to the same line as the one in the living room.

Then there are two options.

1. This is all the same line, the line that goes into your current modem/router, the second plug in the living room and the plug in your bedroom. Then you can't use that line to transmit any extra data.

2. The second plug in the living room is connected with your bedroom plug, but the plug which the modem/router hookes up with is not connected to that same line. Then if there are 4 wires between the living room and your bedroom you are in luck. An internet cable really only needs 4 wires, which is what the RJ11 plugs (I am assuming here that the plugs in the living room and your bedroom are RJ11 plugs, look them up if your not sure) are. So you can create or have someone create two cables with RJ11 on one end and RJ45 on the second end. Hook one of them up to the modem/router and to the living room plug the other one would then go to your bedroom plug and to your computer. Voila, you got fast internet. On the other hand, if there are only two wires connected between your bedroom and living room, your out of luck.

So basically to use the plug in your bedroom you need four wires in that plug that connects to the empty plug in the living room and those wires can not be connected to the plug that your router connects to.

A second solution to get net to your bedroom is using the electricity wiring. See for example this cnet article for options of makes and modules to do this.

I am trying to wire two Ethernet jack ports into a trailer outback so two computers can plug into the wall at the same time. I have one long CAT6 cable with an rj45 connector on one end that is plugged into the router in the main house, and loose wires on the other end.

OK.

Do I need to wire two jacks into the other side of this one cable and plug it into two ports in the router?

Short answer: yes.

Longer answer:

CAT6 (and cat 5 and ..) cables have 8 wires. 4 of those for 10/100Mbit Ethernet. All eight are used for gigabit speeds.

This means that you can split one 8 wire cable into two sets on both ends and run two 100mbit connections over it. You can do this either by creative wiring both ends, or you can use existing plugs similar to these:

This will effectively wire your network like this:

This is likely not officially supported, but it works well enough.

Note that you cannot run gigabit speeds over these cables anymore. You no longer have the 8 wires per connection needed for that.
If you have any electrical power in the trailer (and I presume you do, if for nothing than to run the computers off), then the proper answer becomes a 'do not do this, but a switch instead'. 4 port switches are dirt cheap, do not limit your speed down to 100Mbit and will save you a lot of time.

If you do no have power and are running one or more laptops on battery, then consider these cheap pre-made plugs over building your own cable. Keeping things standard is a good thing.