The box, I assume network interface box, may not exist depending on the age of your installation. It may also be inside your house rather than outside.
In addition DSL may not be available in your area. It has distance requirements (you may need to be within 3 miles of service) that your house may not meet. In addition it will most likely require a NIC card to be added to your computer.
In general, CAT5 most likely isn't rated for outside use infact the jacket is permeable. Make sure the cable you buy is rated for outside use, though I am not sure it exists. Also protect it from the sun, ultraviolet light. This will break down the jacket over time.
I don't really recommend running a telephone line outside when said telephone line is for a computer. You could very well fry a NIC, modem or motherboard if the line isn't protected.
In addition CAT5 is very sensitive to breaks. Do not make 90 degree turns and don't staple it under pressure.
How about cable modem? Either way you may want to have a professional install it if you are not up to installing the cable inside the house.