What I did was to work with a bunch of test papers, with a notebook to write down vocabulary I wasn't familiar with. I'd end up with a list of study words for each level.
You can get practice materials by googling something like 漢検 無料 プリント plus the level (10級 and work your way up) , there are also Nintendo DS games and phone apps. I went with the DS route because I could get them cheap off Ebay at the time, it doesn't matter as long as there's lots of practice questions.
What the lower levels taught me was there were a lot of words that a primary school child would be expected to know that I didn't. I thought I 'knew' all my basic kanji, I'd just whip through the first few levels till it got hard. That was a lesson in itself. It quickly became an exercise in vocabulary learning first and in kanji learning second, but I don't think that was a bad thing. I credit it with getting me to the critical kanji/vocab level that allows you to actually start reading.
I also used another DS game called 'nazotte oboeru' which is similar in concept - good fun as it has themed sections in addition to working your way through the levels (an entire section of food-themed kanji quizzes, this is very fun if you are me).
Kanken practice questions generally have fairly simple sentences to give context so you know which word they're looking for. I don't remember running into any grammar issues (I was probably low-intermediate at the time on that front).
Another potential, if you would prefer to start with something aimed at foreign learners:
2001.Kanji.Odyssey
I haven't used this fully but I had a play on some of their free materials. They provide common words and example sentences (
sample ). I believe there are some tests also.
You will have to experiment with learning styles - some people would take this material and SRS it, I feel my brain turning off after about five seconds of flashcards, but quiz-style stuff like kanken works well for me. I often try to combine more than one sense - so when doing the reading I would read aloud the full sentence, if doing writing I would say the word as I wrote it.
It doesn't really matter which method you choose in the long run, as long as you keep at it. Study every day, even if only for a short time, and you will see your skills improve. Don't neglect your grammar, either! Your aim should be to get to a point where you can read.