I don't want to necessarily dispute some of the interesting posts here (and I think I will take a look at Leangains), but I tend to think the whole "gain muscle while losing fat" thing is actually a mis-statement of the process, particularly when people have a lot of weight to lose.
More often, when one reduces size, muscles simply become more defined. So they appear to have gotten bigger, when in actuality, they have gotten smaller.
Furthermore, significantly overweight and obese people tend to lose muscle mass as they lose weight, since they don't need as much muscle to carry that weight around anymore.
Unless one is a competitive bodybuilder, worrying about trying to do both at the same time will likely be counter-productive (so I'm echoing what Prof. PaleoAP says, to some extent). In all likelihood, if you are very overweight, losing weight and just getting to the gym will make you seem and appear stronger - and you might get some strength gains as you start to use muscles in different ways. But the overall thrust should be for one or the other.
But as others also noted, a lot depends both on what you do and what your genes have to say about things!
I don't understand why lifting twice per week is boring.
Here is my argument for 2 days per week. If you are training balls out with compound movements with max effort and going to failure then it takes time for those affected muscles to heal. Personally after a good workout of squats I don't feel like doing much lower body for a couple of days. If you train like that too many days in a week you won't see many gains as you need to recover from an ass kicking.
Destroy muscles and let them recover in an optimal manner so they will heal up faster and bigger to take on more punishment the next round. Do this over and over again and then you will gain muscle.