Turn bmx bike to 3 gear

You can buy basic 3 speed freewheels on eBay. With that and a claw-style derailleur you should be able to set up a 3 speed system. You might want to just get a different bike instead though.

The bicycle in question has single-speed rear hub and aluminum frame:

Let's consider your options.

1. There is no space to fit any sort of a cassette or a multispeed freewheel to provide more than a single gear at the drive side (but see below). You might be able to find a two-speed freewheel like the one on the picture below. Even if it fits the hub and frame (which is unlikely), to change between the gears you will have to unscrew the wheel and manually reroute the chain. It would be problematic to weld on a derailleur hanger to have a place to attach a derailleur to avoid that.

2. Adding a front derailleur and keeping a single rear cog could be an alternative, and there are triple MTB front rings to offer three speeds. That would mean changing the crankset and the bottom bracket (let's assume it is possible to find a new one compatible with the frame, which is unlikely). To control the chain, a front derailleur has to be installed (should be possible with some luck). To pick up chain slack, a chain tensioner has to be installed at the rear (should be relatively easy as tensioners are often designed for single-speed bicycles). Frame clearance at the bottom bracket or chainstay might limit your options of which front rings could be installed, and whether or not chain would rub against the frame (which would be not OK at all).

3. Changing the rear wheel to one with an internal gear hub. There exist hub variants from 3 to 11 speeds. The hub width must match the dimensions of the existing hub, and of course a handlebar shifter, shifter cable/housing and anti-rotational washers are needed.

4. The bicycle has a flip-flop hub, meaning two different cogs can be installed on it. Unscrewing and flipping the wheel will allow to chose one of two gear combinations. It is impossible to have cogs' teeth difference too big (max two half-links), unless a chain tensioner is installed to pick up the chain slack.

The integrated rear gearhub seems to be the only practical option for converting it into a "real" multi-speed bike. This option's applicability heavily depends on the rear hub spacing, as gear hubs do not come in all possible hub widths. It is unsafe to attempt to alter your frame's spacing to fit a hub of a different width.

Two cogs at the flip-flop hub is the easiest option. Other options can be truly considered as kludges.