How to detect curve in chart grid?
Conceptually, I would go for the following steps:


Perform thresholding of the image to create black/white image.
Detect the grid in the image. How you do this depends on the image properties. For example, will the grid lines always be horizontally and vertically or can the grid also be rotated? If it's always horizontal, you can simply add up the pixel values in all rows or in all columns, and you will see the grid lines in the sums. Also, you can detect the grid lines using Hough Transform.
Remove the grid lines from image, e.g. by setting all pixels that belong to the grid to 0. 
Use dilation to close the gaps in the curve, where the grid lines were removed.
check, where the curve crosses the grid lines by checking, where the grid image and the dilated image are both 1.