The first step is to get a copy of Lynn Pectal's Designing and Painting for the Stage and follow the directions for starch sizing the scrim. It involves HOT boiling ARGO laundry starch, not the kind that comes in a bottle or spray can. The starch stabilizes the scrim as well as sealing it so the paint/dye does not have a tendency to "bleed". Even if the starch fills some of the scrim openings, it is translucent to transparent. The starch size is the best method of sizing a scrim.
For the painting, dye is still the absolute best but in today's world of chemical/safety responsibility most traditional scenic dyes are rarely if at all used anymore. Rosco supersat mixed in a dye like consistency or I believe Rose Brand has a new dye product out, are possibly your best bet for painting.
The method I learned and used for years and which I still tend to think is, if not the best, is "one of the best" methods of scrim painting.
If the picture is not highly detailed or if you do not have access to a high power projector or a way to project on to your paint floor, I would suggest the following:
First layout bogus or butcher's paper on the paint floor. Staple or tape it down so it is stable. Draw your picture on the paper, starting with light charcoal lines and finishing with dark magic marker lines for the finished cartoon. The next step has two different schools of thought. 1. Fasten the scrim down directly over the cartoon. OR 2. lay clear plastic over the cartoon and then fasten the scrim down on top of this. The first lets you see the cartoon the best and has the least tendency to pool or bleed. The second prevents any of the paper from sticking to the back of the scrim, needing to be peeled off later, but one has to be careful not to work too wet. The main thing is that you can see your cartoon and the scrim is fastened down in a true right angle square or rectangle. It should be smooth and slightly taut, but NOT stretched. Next comes the starch coat, let dry overnight, especially if the humidity is high, you want the starch to be VERY dry when you start the paint job. You want it to be a lot like paint by the numbers in that the color needs to be right and in the right place the first time. You can't over paint or paint out an error. You can blend adjacent colors while wet or damp, wet blending is fine as is scumble or other wet blend techniques. Over painting techniques like sponging, rag rolling, and spattering tend not to read as strong as when painting on an opaque drop or flat. Final line work, hard shadows etc. need to be generally darker and heavier than you think.
The trick to painting scrim is to make sure the paint doesn't stay in the 'holes' between the scrim threads. There are several ways to do this depending on if you are working on a vertical or horizontal paint surface. If you're doing it on the floor, lay down bogus paper first to help absorb the extra paint then stretch out the scrim. For the brick pattern you are going to have to take a LOT of extra care in the stretching process to make sure all the weave of the scrim is absolutely straight. Scrim doesn't need to be stretched too tight - just firmly. Once this is done, paint with a watery paint mix. After you get paint on the scrim, but before it totally dries, use one of those cheap, soft-bristle, house-hold brooms to brush the excess paint out of the scrim's 'holes'. This brushing technique requires that you only work with one color of paint at a time. You may need a smaller brush to work the paint for the mortar lines in the brick wall to keep it from spreading into the brick areas. Or you can use towels/rags to mop up excess paint for the detail areas. If some of the 'holes' get plugged up, it won't affect the scrim effect too much - you just want to keep the majority of them open.
If the Scrim is properly stretched, straight lines should not be a problem. You should put some bogus paper between the scrim and the floor before painting. The paint will work its way through all those tiny holes in the scrim and the paper will help to absorb the excess. Even though there is less material it seems it takes more paint than a regular drop. Lay out the scrim in charcoal first using your straight edge, or chalk line, then paint with a brush. A Hudson sprayer will work nicely for the base coat. As for the bricks, I would be more inclined to use a stamp rather than a stencil. Stamps can be easily made, and used a number of times to build up a texture. I also find stamps to be faster than stencils. If you go the stamp
route, experiment on something other than the scrim so you can get the right saturation of the stamp, when you get the feel stamp away.
I believe the real secret to scrim if you are going to use paint is the consistency of the paint. Thin, thin, thin. We had a large scrim professionally painted and the paint was too thick. Too thick means that after folding the scrim and storing it, the fold lines are a problem when rehanging the following year, or whenever. Thinner paint is better.