One method, like in this YouTube video, would be to use heat bond carpet seaming tape. Pro carpet installers use it with a special iron to seam carpet together, but you can use a clothes iron since you will be working on the back side. This would give a strong bond but you will still need a rug pad to keep it from sliding around. (The heat seal tape has a paper backing that is not removable and will be slippery on smooth flooring)
You could use double sided carpet tape in conjunction with a cheap non-slip rug pad. (or not so cheap alternative Pad)
Lay out your samples in the pattern you want, then flip each over so the back side it up. Use the tape to tape all of the seams and put a border of tape all around the perimeter. Place the tape just in from the outside of the outer borders, about a 1/4 inch in. Put two tape strips across, perpendicular, to each seam for added strength in holding them together.
When dispensing and cutting your tape, leave the protective backing on and put the sticky side down, then press all of the tape down firmly. (Use a rolling pin or a can of tomatoes as a roller to help press it down.)
Unfold the rug pad and then roll it up like a big roll of carpet and place it on one end of the taped together samples so that it will roll out and over the samples as you unroll it. (You can trim it down to just bigger then your rug you are making, Dont worry about cutting it exactly to fit, you can do that after it is taped in place.) Position it and roll it out over the rug to make sure it cover the whole rug with little hanging over all four sides then roll it back being careful not to move it side to side.
Now peel of the protective backing on ALL of the tape and start to unroll the pad onto the Taped rug. Use your roller to press all of the seams and edges.
Now you can trim the pad just inside the outer border, That is why we left that 1/4 inch gap. You can use small scissors or carefully cut with a utility knife so the pad is not visible.