Disclaimer: I'm not a vocal coach, I'm just going to answer based on my own experience. If you are trying to increase your range without a vocal coach, then it is at your own risk.
Firstly, if you can afford it, hire a vocal coach, as that is almost certainly the most effective/safest way to increase your singing range. That is the perfect-world solution.
Having said that, I've been self-training to sing the lead vocal on Alice in Chains (metal band) for the past 3-4 months, as I don't have the money for a vocal coach. When I started, many of the highest notes were outside my comfortable range (some of them still are); however, my voice has been getting stronger and I've definitely seen some improvement in my range over that time.
I haven't been doing a lot of scales (although I do make sure I warm up properly first). Mostly, I've been practicing by singing the songs. I sat down and ranked the songs in order of pitch difficulty and started with the lower ones, which were comfortably within my range. Then, I moved on to some of the ones that were a little bit outside my range and worked my way up gradually, once I became more comfortable with those.
It's a slow process and it shouldn't be rushed. Be very careful and stop/take a break, if you feel any pain or your throat starts feeling sore.
I'm sure that doing scales to practice and increase your range wouldn't hurt. Although, the impression I get personally is that the main thing is to be exposing your voice to those higher notes on a regular basis. If you're doing that then your range should improve gradually over time. I think it's quite similar in a sense to body-building (your voice is a muscle, right?) - it's possible to build muscles by working in a gym doing exercises with a personal trainer, or you can do it by hauling bricks on a building site all day. The end result in both cases is muscles (unless you take it too far and injure yourself). So, I think it's important to try different things and find a method of training that works best for you.
In anticipation of some people not liking this post, because I am advocating trying to increase singing range without a vocal coach: The thing is, it's not a perfect world and many people simply can't afford it. So, for those in that situation, the only realistic choices are to either take the risk of self-training, or not sing. Personally, I'm not happy with the second option and want to give some advice for others who might be in a similar situation.
(I should probably also mention that I did quiet a lot of choral singing at school when I was younger, so I wasn't coming at this completely 'cold')