First of all, only use a publisher if you must to get your product through. Publishers can help a lot (and in my case of Sumo definitely did a lot), but there's a lot of benefits to self-publishing too, and I would urge you to look in that direction first.
Secondly, if you don't have a way to make a professional trailer, I would highly, HIGHLY recommend that you hire someone to do so for you. The trailer is, for all practical purpose, the part of the page that will sell your game the most, so if the trailer is good, the game will do well, but if the trailer is bad, the game won't do well.
Next, don't be discouraged by the lack of votes after the first couple days. The graph for every game on Greenlight absolutely plummets after day 2, since it goes off of the "Recently Added" list. This doesn't mean that people aren't as interested in your game - it means that you're to the point to start marketing the game yourself! :)
Add a "Check out this game on Greenlight" sort of button to your game. Then, pursue distribution outlets. I would highly recommend trying to get your game into bundles, which can be frustrating to try to do, but is very worthwhile when it works out. Putting your game on GameJolt, itch.io, and IndieGameStand can help a lot too, but bundles seem to help more-so.
Fifth, and most important - ignore all of the negative comments, unless they're from someone that legitimately wants to see the game work out on Steam. Greenlight is full of trolls, and it's really disheartening to see people beat a game up. But it happens to literally every game on Greenlight, and does not actually reflect the quality of your game, which is really important to remember! :)
Sixth - add some pictures to the description of the page itself. Mine aren't very good, but you can get an idea of what I mean by looking at my Greenlight page for Rubber Ducky:
Seventh - run promotions with giveaways. A good way to get pretty consistent votes is by running giveaways (even on random Steam keys you have lying around) on sites like SteamGifts.com. Add a small note in the description about your Greenlight game, and you're pretty good to go!
Eighth - be ready for the long haul. Some games take time to get Greenlit. Rubber Ducky was in Greenlight for way over a year, I think, but it eventually got in. That's the very, very long end of the spectrum, but if a game doesn't get in after a couple months, it's not just dead in the water. be persistent, and eventually things will work out.
Some very helpful stats:
If your game is at 30% yes vote, you are doing great. If it is below that, you still definitely have a good chance. Rubber Ducky was Greenlit at 21% yes vote - it just took a while.
Sumo rose from a low (I think it was 19% yes vote) to somewhere around 40% after I did some better marketing with bundles.
If you're like me, you tend to focus on the negative comments coming through on the page. That said, often, there's not nearly as many negative comments as it seems there is.