You can do pretty much any stitchwork by hand, it only takes longer than by machine. Anyway, I doubt, that for sewing yourself into tights, a machine would be useful.
Overlock stitch is used to prevent the open end of fabric from fraying. It is also used on lycra seams, to gather the fabric around the seams and make thinner (but harder) seams. I know, because I had to repair leggings a while ago. You simply move the needle through the fabric and back, then almost pull tight, but lead the thread through the loop that is formed, then move on. Machines do that in a different way, but this method works well and looks the same.
There is a simple stitch, that should work, no idea, how it is called, but I am not an expert. It is 1 step forward - underneath the fabric and up again, then 1/2 a step back, then repeat. This creates a strong seam and if you try to rip it apart, you will probably damage your tights.
I found a website about sewing, this stitch is called backstitch and here is a tutorial:
http://sewing.about.com/od/techniques/ss...itch_2.htm