Build a scale without computer
Flashing

Once you upload your sketch to an Arduino, it does not need to be connected to a computer anymore; the Arduino IDE uploads the sketch into Flash of the Arduino, which will stay there until overwritten again, even when the Arduino is powered off and not connected to any power source. The uploading process is called Flashing (done by the Arduino IDE normally).

USB power

For powering, instead of using the USB plug to the computer, use a socket adapter with a USB plug, same as a telephone/USB phone charger. This will have the same behavior as connecting it to a computer, except that flashing is not possible.

Battery power

(from comment below)

You can simply connect the battery to the Vin, because the Arduino already has a voltage regulator (which works recommended from 7 to 12 V and absolute maximum rating of 20 V). The voltage output from e.g. the 5V or GPIO pins will be (approximately) 5 V. The reason that you the minimum voltage should be 7 V of Vin, is that the internal voltage regulator is not 100% efficient (no voltage regulator is).

So if you connect a battery that is at least 7 V (e.g. a 9 V battery or 5 or 6 AA batteries you are ok). Note that if you use 5 V batteries from 1.5 V, you get 7.5 V so the limit of 7 V will be reached quite soon as the voltage of the batteries drop.