What is the proper way to frame in a room?
You are going to need to pull permits unless you live somewhere that doesn't have any requirements for them.  Since that is probably visible from the road, there is almost no way that code enforcement or one of your neighbors isn't going to see it and report you.

The stairs look like the precast concrete kind and probably are hollow and not attached to your foundation. 

The method you are proposing sounds more like building a deck than building correctly for a house. Most likely, to be legal, you'll need to remove the slab and dig down beneath the frost line and build up with poured concrete or with cinder blocks.

If the slab happens go down around the edges far enough to support the weight, fine, but most likely it doesn't.  Assuming you don't care about being legal, you could build up a cinder block wall to the height level with the bricks on the existing part of the house, setting them back so that you could brick the new parts at a later date.  The bricks from the first place might be enough to recycle. Then build your new walls on top of the cinder block walls. 

The nice thing about doing everything legal is that it will let you remove that fire place and any of the columns. That fireplace almost definitely wouldn't be legal for inside use.