From your question, you seem to have 3 issues:
1. Keeping your back straight
This one shows by you feeling like you're falling over. This problem can be taken care of by doing two things:
Work on your core
Your core (abdomen, waist, lower back) carries your upper body. Strengthen that and you'll have less trouble keeping your back straight. This should already be included in your regular training (situps, crunches, plank, etc.), so just keep at it. Don't cut corners on those.
Practice in front of a mirror
This is something I recommend for everything, like your basic stances, forms, etc. It'll help you get the right posture. I cannot stress enough to do this with mirror from the start, so you get it first time right. Correcting it afterwards, after years of practicing it in a wrong way, will make it a whole lot trickier.
2. Putting your legs into that pose
This one is mainly a matter of flexibility. There are some exercises that can help with this.
Obviously, stretching is an important part. Sean Duggan's post covers the most important stretches for this
Rotations may be a weird term, but I don't know the exact word for it. It comes down to hip rotations and rotations of the part where your leg attaches into it. These include the exercises where you swing your leg upwards, keeping it stretched along the swing, inwards, and outwards (crescent moon kicks). These should actually already be part of your regular training too, so never cut corners on those either! There are other exercises that are great for it too, like going into split, then turning around. Those are more advanced, though, so keep at it with these, first.
3. Staying seated like that
The last issue is staying seated like this. This is done with a combination of the 2 steps above, and simply just sitting like this. If you really can't keep sitting like this for even 30 seconds or so, try against a wall, first. The extra support you get, will compensate for the core that has to be built up, and may help with keeping your back straight as well. Move away from the wall eventually, so you can sit by yourself, without the support, and just keep at it. In the end, you'll be able to sit in lotus position for extended times with no problems.