If you want the human race to vanish from this planet, you'll have to make them vanish all. Otherwise, there will always be the problem that the remaining people would make the whole game start all over again. (Natural) diseases or hunger won't be sufficient.
I see four (virtual) strategies which might be successful
1. Get them all at the same time
2. Destroy their means of existence
3. Block their offspring.
§ 1 - Could always be reached by a cosmic catrastophe, such as a meteorite impact, but I think, this is not really the topic of this thread.
I think that we are technically already able to blow ourselves all up. On the other hand, it is not the purpose of a war to kill oneself, too, so this could only happen by unprofessional handling of this tool.
On the microbiological level, I think it is/will be also possible to create a virus more efficient than the pest, which let anyhow two thirds of the European population alive in the Middle Age.
How probable is this to happen within the next 1,000 years ? We'll see.
§ 2 - Man is very versatile concerning his food stock and the physical conditions of life. So, it won't be sufficient - as it is for other species - to clear some square miles of rainforest to exterminate him. As we know, a climate change can be induced by the human race, but it must be a really extreme change in order to make end to any agriculture and to anything else to eat or to drink. I think that our species will only leave Earth as one of the last ones.
Soil erosion is potentially also effective to make an end to agriclture and to wildlife, but in such a case, the last survivants will be smart enough to live on the residues.
To make vanish or to poison all drinking water is obviously very difficult to realize as long as the cycle of evaporation and rainfall remains intact. I myself at least wouldn't know how to make it collapse.
§ 3 - One theory of the extermination of the mammouth is that the ice age people only hunted their calves, with a weight below one ton. In any case, attacking procreation is a very effective virtual stategy to make end to a species. What could make us stop to have children or what could kill them systematically or make them infertile ? One reason is that it could be "out of fashion" to have kids - in many indrustial countries the birth rate is much lower than 2 children per woman, so below the rate of self-reproduction. However, I doubt that this could really be extrapolated down to zero - before, some people could change their minds.
What else could stop the procreation ? Poisons, diseases, socially or educationally induced asexuality, food ?
As a conclusion, I dare the prediction that there will be a big crisis for our species, but not (yet) the real end.