Disagreeing politely Suppose I disagree with the methods and/or conclusions of a paper published in a scientific journal, and I produce some new results that refute the claims in this paper. What would be the best way to publish my contradictory results in a way that challenges the original paper, but also shows respect for the authors of the previous paper? Should I explicitly mention the positive merits of that paper before explaining why I think that the conclusions are wrong? Or would it be implicitly assumed that I disagree only with the previous paper's scientific conclusions, and I have no negative feelings towards its author(s)?
I would like to ask a counter question; what it the main motivation in you wanting to publish your contradicting results and your disagreement with the authors of that one paper?

I think the main motivation of a publication should be to add something to the literature, not the discuss results back and forth. That's what conferences are for.

Without knowing the specifics of your particular case at hand I see the following possibilities:

Experiment presented is not the best way to test the hypothesis

If your reasons for refuting the article originates from the approach presented in the paper, then you are merely proposing an alternative approach to  solve the same problem. 

That's the easiest scenario, in my humble opinion, since you can easily refer to Smith et al. and say that your approach is superior/more accurate compared to theirs because X,Y,Z. That would essentially be a standard "original research" paper. 

Incorrect results

if you have tried to replicate the experiment by Smith et al. and cannot come up with the same results, then you can adopt the following steps in that order:


Redoing your experiment, once more, retracing your steps to make sure you have done exactly as explained by the authors.
Contacting the authors to tell them that you cannot replicate their results and whether or not there might be a key step that you might have overlooked. This is a sign of good faith for your fellow researchers. 
In the case where despite all your efforts there is something just not right, you can try contacting the editor with evidence on contradicting results, which may lead to a correspondence type paper. This approach might come across as a more or less accusatory approach however. 


Misleading conclusions of results

If you think the specifics of the experiment (or equivalent) are solid but the conclusions drawn and presented are misleading, you can write an opinion piece, or a commentary and send it to the editor. 

Be sure that the discussion (i.e. whose conclusions is more accurate) is actually relevant to the general audience in that field. Otherwise it's just scientific banter which can be carried out by email.