How to decide on the order of authorship for this collaboration? I'm in collaboration with another researcher in the field of linguistics, she wants to try machine learning algorithms/analysis on her dataset. The dataset is not published before, and the contribution of this work is to use ML to better analyze linguistic theories and to support/challenge them with this new empirical evidence (ML).

So, my role here was to try appropriate ML tools based on the goal she described, and also I designed a new ML algorithm to serve the purpose. In addition, I perform the experiments, numerical evaluations, preparing the figures. 

But since the main concept and the aim is fundamentally related to the linguistic field, she is in charge of writing down the main structure of the paper (Abstract, Introduction, the purpose, contribution). 
But, definitely, I'd write the ML parts and where I need to provide quantitative conclusions and also explanations regarding the new algorithm I designed and ML approaches that were used, and how to interpret the results. 

Well, we are aiming for a conference in the area of computational linguistics, and it is not clear for me who should be the first author of this contribution.
Also, I do not know if it is possible to claim equal authorship for such cases, and whether it'd be considered as the 1st authorship in my CV?

BTW, she is a post-doc, and I'm an almost finished Ph.D.!  ;-)

Update: Since the value of this publication (if being accepted) is important to both of us regarding our future proposals and job-search, I personally prefer an equal authorship option, if possible!
But I'm not sure if possible to show that in the CV!
Here is a thought.  (Perhaps it is naive.  I haven't seen your paper, and I know nothing about the publication culture in computational linguistics.)  

Could you perhaps write two papers, with you appearing as first author as one and your collaborator on the other?  This would settle the issue symmetrically.  Whether it is a good idea depends more on how "bi-disciplinary" your work is.  By which I mean: is your work coming from two different camps, linguistics and machine learning?  If not, splitting is a bad idea.  If so, if you write one paper "splitting it down the middle" is there a ready-made audience who will appreciate the synthesis of both camps into a single paper?  If so, splitting is a bad idea.  But if not, you might find that your work reaches more people if you appeal separately to the linguistics and machine learning communities.

Again, just a thought.