I am building an ambilight. So I bought a 3 pin LED Strip which I power with the USB hub from my TV. Next step is to connect it to the Arduino Nano. Is it OK when I only use the dataport from the Aduino and the power only from the television?
I think it is not a good idea to power the LED strip via USB.
Normally USB delivers a maximum of mA (see below). Assuming you have 30 LEDs (minimum for a normal LED strip), and each LED gives 40 mA (maximum), this is already more than what most USB ports can deliver. This can ruin your USB port.
For the Arduino Nano you can use the 5 V, and you should connect all GNDs together.
But you will need a decent adapter anyway for the LED strip, so than you can power your Arduino Nano from the adapter you use for the LED strip.
According to Wikipedia USB power
Low-power devices (such as a typical USB keyboard) may draw at most 1 unit load (1 unit load is 100 mA for USB devices up to USB 2.0, while USB 3.0 defines a unit load as 150 mA), and all devices must act as Low-power devices when starting out as unconfigured.
High-power devices (such as a typical 2.5-inch USB Hard Drive) draw at least 1 unit load and at most 5 unit loads (500 mA) for devices up to USB 2.0 or 6 unit loads (900 mA) for SuperSpeed devices.
Can you follow this? Connect all GNDs. TV 5V can power the Arduino, connect to the 5V/VCC pin and not the Vin pin. 5V power supply only powers the LED strip. No reason it couldn't power the Arduino as well tho, in which case there would be no need to connect to connect anything to the TV.
PC USB ports typically can only supply 500mA, only enough for 8 RGB LEDs at full white. I doubt your TV can do much more.