Okay, so the basics are out of the way, let's talk about how to really gain those views and watchers.
The goal you want to aim for is to be featured on the first 10 spots or so of the popular 24 hours listing. This is because by default, users who are logged into deviantArt defaults this page as their home page. So whether they're trying to go to their profile or check out their notifications they're going to see the top 10 deviations featured. Essentially, despite the fact that the actual 'front' page of deviantArt is a different page that they put a lot more work towards, the popular 24 hour listing became the new place where most people see the top deviations. That means that if you can get your newest deviation into that top 10, for up to 22 hours or so almost every deviant who is going to their page will end up seeing your deviation. So this is pretty much the holy grail of trying to get new watchers.
However, you'll notice that its usually dominated by people who have huge watcherbases, ie, if you're already popular, you'll get even more popular, but if you're not popular, you can't really break in. Of course, its not impossible to do so, because you'll often see deviations on there from deviants with very few watchers and relatively little views/favs compared to the popular deviations. Often that one deviations will end up having vastly more views than anything else in their gallery. So how does that happen?
Its a combination playing the categories system, having a dedicated fanbase, and a bit of luck.
DeviantArt has system where they don't want art of the same type in their popular 24 hours listing, so generally they only take the top from from each category. For example, the #1 deviation in MangaAnime/Digital Media/Drawings might be #3 on overall list, but #2 could be all the way down at #33, even if it has more views/favs than the deviations ahead of it. So you don't need to compete against other popular artists if you go for categories that are not as popular.
Not all categories are created equal. DA has been trying to promote less popular categories. For example, if you try to put your art in popular categories like Fanart or Manga/Anime, its pretty much impossible to front page because you're competing against people with 60,000+ watchers. However, there are a number of unpopular categories like Fashion Design where its a lot easier to take #1 in that slot and appear in the front page. If you're absolutely shameless you can put art in completely unrelated categories and just apologize later. That's actually happened to a few deviants who were really blatant in abusing the system.
The categories are not transparent either, for example, Manga & Anime seems to be its own category, so all the ones ranked below it are generally put in the same category. However, 'abstract' is another category that appears often in the top, but 'abstract' is listed under multiple top level categories, such as Digital Art/Mixed Media/Abstract, or Traditional Art/Drawings/Abstract.
The rankings within the categories themselves depend both on the rate you get views/favs per hour along with the fav to view ratio. I don't know the exact formula, but having a higher favorite to view ratio will allow you to beat out pics with high views but much lower ratio. Of course the absolute fav count plays a significant role too. Generally though, even if you choose a less competitive category you probably still need around 50 favs in your first hour.
So the method is to slowly grind your way until you have enough watchers so that in the first hour, enough people will favorite your work to get you to #1 in your sub category, and thus give you a chance to get into the top 10 listing in all categories.
Since I've started my account about a year ago, I was able to front page 3 times. The first time was when I had around 2.5k watchers, and I had a piece that fit in Traditional/Mixed Media, which is a very uncompetitive category. However, even though I dominated that category my overall number of views/favs was only able to put me in the #17 slot for about a few hours. The 2nd time was another Traditional/Mixed Media piece I posted when I had about 6000 watchers. That one also only front paged for a very short while, and I don't think it broke the top 20. However each time I did get a huge boost of watchers, and until recently they were my two most popular pieces.
The third time was this past Monday, where I posted my latest piece in Digital/Mixed Media/Abstract. This isn't that competitive of a category but one of the really big artists routinely posts stuff there. I made sure not to publish it when that artist published his. This time I had 13.5k or so watchers and was able to stay at the #4 spot for about 17 hours. It got me a TON of views and 166 new watchers in the 24 hour period it was up. So yeah, front paging is by far the most effective way to gain a larger fan base.
Another effect of people favoriting your art is that they show up on their page under their favorites. The more people that favorite your stuff the more it spreads, especially if someone who already has a large fanbase favorites your art and it shows up under their favorites.