Thats not going to give you internet by connecting it from your computer to the router because that's a phone line set up. Any time you see a connecter with just red and green wires then that's whats used for connecting phones or it could be used for DSL.
The fact that it has a Rj45 connector makes me think DSL.
To make full use of the Rj45 connector and have it connect your computer to the router to get an internet connection there has to be 8 wires in that connector. Not all 8 wires are used but ethernet cable is an 8 wire cable.
the 2-wire (also known as single pair) type is usually used in regular phone systems. in contrast to that, ethernet (or network - usually through cat5 or cat5e) type cables use a 2 pair, or just 4 wires for transmission while the actual cable itself has 8 wires. in theory, you could run phone and ethernet through the single cat5 wire since there are 2 pairs being unused.
also, you can hook phone (rj-11) or ethernet (rj-45) tips to either type of wire.
it really looks like a possible VOIP phone type wire with the rj-45 tip.