How do you write Katherine in Chinese?

You don't. Mandarin (chinese) does not have a phonetic alphabet.

Not all English (or foreign) names have an equivalent in the Chinese Language.

Just use the English reading when in doubt, otherwise the Chinese phonetic translation may lead to unintended humourous results.

However, 'Catherine' or 'Katherine' is commonly translated into 凯萨琳 kai(3) se(4) lin(2) in the Chinese language. 'Kathy' would be 凯西 kǎi xī, 'Kate' would be 凯特 kai(3) te(4).