I assume what you want to prepare is normal ("physiological") saline solution with 0.9 % mass concentration of NaCl, because that's what is used in such applications.
Please note that I'm not a medical professional and this is not a medical advice. Anyone interested should absolutely do their own additional research and never blindly trust any individual on the Internet.
Formula for this is:
- measure 9 g of sterile NaCl;
- dissolve it in a volume of distilled and sterile water below 1000 mL, let's say 700 mL;
- make up to 1000 mL of volume with distilled and sterile water.
Please note that the reason of the formula not just saying to dissolve 9 g of NaCl in 1000 mL of distilled water is because doing so would increase the total solution volume a little beyond 1000 mL and thus result in imprecise (a little lower than expected) concentration of NaCl.
However, I think this level of precision isn't as important in your case of external use - if you just dissolve 9 g of NaCl in 1000 mL of water, it will be precise enough for this application.
A source for the formula is a relevant article at Wikipedia, also available in your native language.
What is more, please note that the formula mentions that the water is distilled and sterile; and that pure NaCl is used.
Water from tap contains dissolved minerals. Some of them (calcium and magnesium bicarbonates), which are referred to as "carbonate hardness", are removed from the water during boiling - it causes them to precipitate as limescale, thus removing them from the consideration. However, other dissolved minerals (referred to as "permanent hardness") don't form insoluble precipitates during boiling - these are going to stay in the water, increasing the total amount of dissolved minerals in the resulting saline solution, and thus increasing its osmotic pressure beyond the intended value. Fine precision is critically important in case of intravenous use of saline solution, but in your case of external use I'd expect that it shouldn't make a difference - the amount of dissolved minerals, even in extremely hard water, is only a fraction of the amount of NaCl in the formula, after all. I'm also simplifying it a little - it's more complicated; dissolved minerals don't exactly exist in water as minerals, but as ions; also, boiling the water doesn't necessarily remove all of the carbonate hardness.
Boiling the water doesn't technically sterilize it because bacterial spores could still survive this - one would need to use UVC light, specialized semi-permeable membrane filtration or an autoclave (which uses pressurized water steam with temperature over 100 °C for extended period of time) to reliably get rid of them and actually sterilize things. It is incredibly important that the saline is sterile if it is to be used intravenously or stored for an extended period of time, but I think it shouldn't be a concern in your case because you are going for an external and immediate use - for that reason, boiling should be more than enough. However, I'm mentioning it anyway as a safety precaution - I'd completely advise against intravenous use of home-made fluids.
Also, please note that table salt doesn't necessarily equal pure NaCl - it often contains a small amount of anti-caking agents and sometimes small amounts of iodine salts (iodides or iodates) if you were to use iodized salt. It could also contain a small amount of dextrose (a simple sugar, also known as D-glucose or just glucose) to stabilize the iodine salts - dextrose is a reducing agent and it inhibits oxidation of the iodine salts by atmospheric oxygen. In case of some countries, table salt could be also fortified with trace amounts of fluoride salts. Again, it shouldn't be much of a concern in case of external use.
Anti-caking agents are, as far as I know, relatively inert and water-insoluble substances - and, being in table salt, they are food-grade, after all - but they may cause some mechanical irritation of eyes if present in this irrigation solution. In your case, if you notice there is some visible sediment floating, you could wait a while after preparing the solution, so that it could settle on the bottom and you could decant the clear solution afterwards. Using a coffee filter is also a reasonable option.
Iodine compounds' concentration in table salt is incredibly small and iodine is a vital trace micronutritient after all. I can't assure you it would be completely neutral, but I'd personally say it should be safe for external use if you were to use iodized salt in your formulation.
In conclusion, I've added a lot of safety disclaimers for formalism; I always prefer to err on the side of being a bit paranoid with safety precautions. However, I'd expect that in practice the potential risks of using home-made saline in your case for cleaning should be outweighed by risks of doing nothing and waiting until tomorrow with cleaning the dogs' eyes.
I hope your dog gets well soon.