Generating A Code
First, you need to go to the code generator, which is here. There are a bunch of links to it across the site and elsewhere, so don't worry about bookmarking it or remembering it.

The page you'll come to will all be in Japanese. Ignore it. Click on the one link on the page.

The next page will also be in Japanese. Don't worry about the red text unless it clearly says PROXY, which effectively means that your IP is banned and you won't be able to vote with that IP. However, if you get the message 'コード発行制限中。。。しばらく待って。', then you're in the majority who will find it very hard to ever get a code unless the organisers relax voting restrictions.

You now need to wait for your code to generate (loading the previous page sends the request but doesn't generate it immediately). Your code will generate at the next but one XX:55. For instance, if you loaded the page at 8:05, the code will generate at 9:55. 10:50 to 11:55. 6:58 to 8:55 and so on.

Refresh (or return to) the page after that time and your code will have appeared, looking something like [[AST104-uReKqvqb-YY]]-00026.
Posting On 2ch
With your code at hand, you need to find the right thread to post in. First off, head to the voting board here here.

The current Saimoe thread will almost always be right near the top, but the surefire way of finding the thread is to search for アニメ最萌トーナメント2013 and go with the result with the highest Round number. You can use this thread finder to make the process easier.

To vote, make a post in that thread containing both your voting code (including the [[]] and unbracketed number) and the names of the characters [in Japanese] you want to vote for. You'll find all of the names in the right format here - make sure you include the <<>> around each name. Along with these you'll need to post some support for the characters you're voting for. Something like 'I love my waifu' at the end of your post will suffice if it's too much :effort: to do anything more.

You may only vote for up to one girl per match. If there are multiple matches that day, then there is no problem with voting for only some of the matches. If you post two or more characters from a single match then your vote will be disqualified. In the first preliminaries you may vote for up to 10 characters per group, and in the second preliminaries you may vote for up to 12 characters per group.

When you click to post (using the only button immediately next to the box), it may lead you to a page other than back to the voting board. If it has a grey background and there are buttons at the bottom then just click the button and you're done. If it has a white background, doesn't automatically redirect you in a few seconds and is obviously an error then your IP has been banned from 2ch[*]. In this case, go here instead and repeat the process (posting on both if you can at best gives you one vote and at worst gets your vote discounted). If you're banned from there as well, then you'll have to find a friend to post your votes for you.

*Other errors you might generate are your post being too long, and if you're posting in a full thread by mistake. Translate the error message if you're unsure.