daisy chain more outlets; wiring
I'm not sure I understand your proposed installation, but this is what you do, assuming the outlet is not switched in any way. I understand the two pairs of wire are both physically connected to the outlet. This sounds like a North American installation, I am assuming such, but instructions should apply to other areas, just that what each color does may vary. One pair is in coming power, the other continues on to feed other outlets or lights.

Remove all the wires from the outlet. Always connect all blacks together and all whites together (and bare grounds together). Each wire nut connection will have the two original wires, the new wire feeding new outlets, and a short pigtail wire. The pigtail is connected to the original outlet. Black wire connects to the yellow or brass colored screw side. The white connects to the silver colored screw side, usually marked as "neutral" (in North America).

I recommend properly binding the wire to the screws instead of using the convenient push-in connectors if you have a choice. IMO, you get a much more reliable connection this way, but the connection must be properly made up.