I have a Godox V860C light and a Godox Cells II remote trigger.

OK, from this, I gather that you have a Canon camera. The Godox V860C is an eTTL-II-capable flash, so if you want to use that capability, it's there. However, the Cells II triggers are manual triggers that do not communicate eTTL information, so with this specific combination you do not have eTTL off-camera.

Does E-TTL work when the flash is off shoe?

Yes, if your triggers, flash, and camera all support it. If, for example, you were to use RadioPoppers, TTL PocketWizards, Yongnuo YN-622C, Phottix Odins, or Pixel Kings with your V860C, you'd have wireless off-camera eTTL.

...does that mean the E-TTL is pointless whenever shooting off-shoe?

Not necessarily, but most shooters prefer to use the M flash mode for off-camera flash, for consistency and precision of control. eTTL is metering based, and is more useful for on-camera bouncing for event shooting (e.g., weddings, parties, etc.) but can change the exposure shot-for-shot when used in an off-camera studio setup, and most off-camera setups take time anyway, so reshooting and adjusting isn't as much of a problem vs. trying to anticipate and snag a one-time-only moment of someone's special day.

I put the flash in manual mode and on full power (1/1). I should be in manual mode, right?

For the most control of flash photography, having both the flash and the camera in manual mode is typically preferred, but not necessary. It depends on what you're trying to accomplish. The main reason folks use the camera M shooting mode for flash is because the camera's meter can only measure the light that's in the scene. Flash isn't in the scene. So you often mentally have to compensate for what the flash will add to the scene, and sometimes that compensation will go past what EC settings let you do. M doesn't have those limits.

Also, using the flash at full power is something you need to do judiciously. It's the easiest way to overheat your flash, as well as requiring the longest recycle time between bursts. Using speedlights off-camera becomes all about conserving the flash's power.

To get the proper exposure, ...

Ok, here's your problem in how you're thinking about flash exposure. There is no one single combination that's "proper exposure" when it comes to flash. With ambient-only exposure, you're probably thinking all you need to do is get the needle to zero. And as long as you can find an iso/aperture/shutter speed combination to get that needle to zero, that's all you have to do.

Flash doesn't work like that. Because a) it doesn't show up on the needle in the camera because it's not in the scene yet, and b) you can have any combination of flash vs. ambient. You can have it be all flash and no ambient (lit figure, black background), or nearly all ambient and very little flash to just lift the shadows (fill) and everything in-between. How you choose to balance your flash against the ambient is completely up to you and the look you want to achieve with the lighting.

The main thing to understand is that the amount of light from the ambient is controlled by iso, aperture, and shutter speed. But the amount of light from the flash is controlled by iso, aperture, subject-to-light distance, and the flash's power setting.

... do I adjust the light until the histogram looks OK?

It's up to you, but understand there's no one "histogram looks OK" point. You still have to choose what you want. It's all going to be about what kind of light and image you want.

Should I do exposure compensation (+1/3 stop? +2/3 stop?) to get the right exposure?

You can try that. However, on Canon cameras, the EC that you do with the iso/aperture/shutter speed is independent of the FEC (flash exposure compensation) that you dial in on the flash to adjust the flash power in eTTL. And if you're shooting in M on the flash, you just choose a power ratio. I tend to start out at 1/4 and see which way I need to go, rather than dinking about with eTTL/FEC.

Think of using eTTL/FEC like shooting in Av mode on the camera with EC. And M on the flash like using M on the camera. M on the flash lets you specifically set the power output of the flash by the ratio of the full power. 1, 1/2, 1/4, etc. And that each of those settings is a full stop (1EV), because it's a doubling/halving of the light.

See also:

• http://strobist.blogspot.com/2006/03/lighting-101-balancing-flash-and.html
• The "EOS flash bible" bit on camera shooting modes: EOS flash photography confusion.