If you have a PNG which has too few pixels (note that ppi settings are irrelevant), then you probably can't do anything that will make it better. Enlarging a low resolution PNG will always make it look worse quality. Probably not what you want to hear. For this reason, raster images are generally not used for logos by graphic designers. You should ask the person who created the logo if they have it available in a vector format, such as SVG, AI or EPS.
If you can't get the vector file, then depending on how many pixels the PNG has, you may be able to do something to fix it. There are a couple of possibilities.
- You could Auto Trace the image using vector software such as Inkscape
(which is free) or Adobe Illustrator
- You could Auto Trace the image using vector software, and then manually
edit the curves and anchors to improve the trace.
- You could just redraw the image manually in vector software, using the Pen Tool (Bézier tool) to trace over the top of the raster image. This option requires quite a bit of skill, and probably isn't practical if you are a novice.
Here's an example of a raster image of a monogram. I show what's possible with Auto Tracing in Inkscape (using Path > Trace Bitmap), and manual redrawing. As you can see, Auto tracing sometimes fills in areas of fine detail, but this might be good enough for your requirements.
Click on image to enlarge