Doxycycline need to convert from 100 mg tablet to equal 1 ml liquid for my cat with upper respiratory infection What is difference?

I'll start from the beginning and say this is a very bad idea - your veterinarian will prescribe the appropriate antibiotics as needed for your cat's infection. This is if your cat even needs antibiotics - most upper respiratory infections in cats are caused by viruses which won't respond to antibiotics anyway.

Second, there is a major difference between 100 mg and 1 ml. The first is a measure of mass while the second is a measure of volume. Neither one is actually a dose, by the way - you need mg/kg to determine a dose of antibiotics. There is also no way to convert a 100 mg tablet to 1 ml of medication because there is no concentration - you need to know mg/ml to make this conversion.