You can't enlarge raster image perfectly but you can trick it a little bit.
For an example:
You have an image with 10x10cm 72 dpi.
1. Print your image with high quality printer & paper.
2. Scan your image in high dpi (e.g. 300dpi).
3. Resize your image in Photoshop (image > image size > uncheck "Resample Image" and change it to 150dpi). Now you've 20x20cm image with 150dpi
4. Refine your image with "Level", "Hue Saturation", etc.
5. Sharpen the image using "high pass":
- Duplicate layer
- Filter > Other > High pass > set 1 or 2px
- Desaturate that layer and change layer blending mode to soft light.
- If you need more sharp, duplicate that layer.
This is work nicely for digital print, and can't work great for offset. It's enough for screen printing
Hope that helps