Efficiently driving 6 LEDs in parallel
You're on the right track with the DCDC converter. Use the 5V and step down, and sense the LED current on the low side. If you choose a regulator that has a 0.6V reference, the loss will be low - you'd use a 5 ohm sense, so the loss would be 72mW at 120mA.

If you want even lower loss, use an op-amp with a lower-value sense resistor to boost the sense voltage up to the reference. That's also an opportunity to apply dimming.

On the other hand, using a purpose-built current-sensing LED driver will use a much lower overhead voltage for sensing as it has the sense amp built in. More about this below.

To get dimmability, you can feed in a PWM chop to the sense voltage to vary the current, or if you have the space, a current sink/source DAC can do that (Maxim makes an I2C part that does this.)

If you're not so fussy about the chop frequency, just use the EN pin of the device.

Here's a cheap DCDC that I've used many times from Diodes Inc. It has direct crosses with a lot of vendors including Richtek, TI and others. It's 2A, more than you need, but very small. https://www.diodes.com/assets/Datasheets/AP3402.pdf This would use 5 ohm sense, or if you insert an op-amp, you could use a lower sense value. I've bought AP3402's (or similar) for as little as 7-10 cents in volume.

This is a purpose-built, quite fancy buck DCDC LED driver with high-side sensing, dimming and other features, also from Diodes (Zetex): https://www.diodes.com/assets/Datasheets/ZXLD1320.pdf It's a complete solution. This part's over a dollar. Much more expensive than the AP3402 + op-amp.

And finally, probably the best solution. Since I seem to like Diodes so much, here's a cheap-and-cheerful LED DCDC with 0.1V reference voltage for low-side sense, and it specifically supports PWM chop: https://www.diodes.com/assets/Datasheets/PAM2804.pdf You'd use a 0.8 ohm sense resistor with this. Very cheap too: 16 cents from Digi-Key if you buy a whole reel.



One problem with LED backlights is matching. If the individual LEDs have small variations in Vf, then when they're driven in parallel there will differences in brightness: the ones with the lower Vf will get the lion's share of the current. To help with this, even with a current mode driver like I'm proposing, keep a small series resistance on each LED cathode (maybe 10-20 ohms) to help compensate for the variance. If they are matched, so much the better, then you can replace these with zero ohms and reduce the losses.

A more sophisticated approach would be to use multiple current mirrors. That's probably more complicated than you want to get into.