Yes, you can do that. You just have to be sure you have enough voltage for the combination of LEDs, and use the right resistor value.
The limiting factor in the booster circuit is generally current. (There's a voltage limit as well, which depends on the type of transistor you're using for the booster, but for typical NPNs it's high enough that it won't be a factor, usually 40V or higher.) When you connect LEDs in series, a single current stream flows through all of them, so the total current that the booster has to handle is the same as the current for a single LED. That means that you can in principle connect infinitely many LEDs in series, and you'll still have the same current as one LED. However, the trick is that you have to increase the voltage to match. Each LED has a "forward voltage" (often written as "Vf") value that you can find in its data sheet, and when you connect multiple LEDs in series, you have to add up all of the single LED voltages to get the total overall voltage. The supply voltage has to be slightly higher than that total. So, for example, if you're connecting 4 LEDs with Vf of 2.2V, the supply voltage will have to be higher than 8.8V. That'll work with your 12V supply. If you have four LEDs with Vf of 3.4V, the total is 13.6V, so 12V isn't enough to power it.
You'll also have to choose the appropriate resistor size for the combination of LEDs. There's a nice online calculator here that will do the math for you:
http://led.linear1.org/led.wiz
That will also show you the wiring diagram for the serial circuit, to double-check that you have the right wiring in mind.
One additional note: You mentioned daisy-chaining the power and ground connections to the LEDs. "Daisy-changing" usually means that you're connecting all of the like connections together - connect 12V to all of the LED (+) sides, and connect the booster port to all of the LED (-) sides. If that's what you had in mind, you're actually talking about a parallel topology for the LEDs, and what I said above doesn't apply to that. In a parallel connection, the current through the booster will be the sum of all of the single LED currents, so the current limit on the transistor is the limiting factor. To check if that's safe, you should take the "forward current" values for all of the LEDs that will be connected to the same booster circuit, add them together, then make sure they're less than the maximum collector current (Ic) for the NPN transistor you're using. The common NPNs like 2N4401 can handle about 400 mA, which is enough for many small (20mA) LEDs, but only enough for one of the large "star" LEDs (those run at about 350mA per channel).