Even though your document is in CMYK, it sounds like your colors have been defined as RGB at some point. The black and grey areas get a color tint when they consist of not just black.
First, be absolutely sure that the document is in CMYK mode (File -> Document color mode)
Second, select a black or grey element, open/go to the color window. If the sliders are anything else than CMYK, select CMYK from the window's hamburger menu. Now that you have the sliders, you can see what the colors really are and adjust them as you like for each element.
As for the blue, hard to say, but if it was RGB at some point, it probably has too much color in it, if the total percentage of color is above 250 %, it usually gets quite dark.
Also, take into account that it is impossible to match 100 % what you have on a screen with print.