For official documents and the like, I write my name as ジェシカ パーカー. This comes out in English as something like 'Jeshika Paakaa' with long vowel sounds in the 'parker' bit.
However, most foreign people who come here at some point choose some
kanji to represent their name.
Kanji, the complex characters imported long ago from China, are used predominantly for names in Japan. Kanji for names tend to be very complex and can have many readings. For example, the kanji for 'west' 西 can be read
nishi,
sai, and
sei. Anyone who has sent me snail mail before might be interested that the 'Sai' in 'Saihaku' means 'west.' Many kanji have far more readings than this.
Following current naming trends in Japan, one could go the route of
ateji and choose kanji based on meaning and arbitrarily assign readings to them. An example here is the word 'Tobacco', which came to Japan many years ago. Tobacco is usually written in hiragana like this: たばこ (tabako). However, in the Meiji era (1868-1912), this was commonly written in kanji as 煙草. If you used the typical readings of these kanji it would come out 'ensou' or 'kemurikusa' or something like that, but these kanji were read as 'tabako' regardless of their true meanings.
However, most parents in Japan still use traditional readings when they choose kanji for names, and most foreigners who choose kanji for their names do the same.
Ideally, one can find kanji for ones name which both have the appropriate sounds as well as a good meaning. For my first name, I need 4 kanji. One to represent 'Ji' 'E' 'Shi' and 'Ka'.
Here are some 'funny' examples I've come up with:
児枝鹿 - Newborn, Branch, Deer
時餌死蚊 - Time, Animal feed, Death, Mosquito
As you can see, choosing based on sound alone could leave one with a ridiculous name.
In English, as per my 'name meaning' mug that currently resides at my parent's house in Florida, my name means 'Grace of God' in Hebrew. Therefore, I wanted to have a name that had the meaning of Grace or something similar in it.
Here are some kanji I like for my name: 慈恵紫花
慈 - Ji - mercy
恵 - E - blessing, grace
紫 - Shi - purple
花 - Ka - flower
I found a keychain with the first character on it. :)
For my last name, I like to use 公園人 (public recreational) park-er or 駐車人 - (car) park-er. The first corresponds to the name's origin in English. The second is for laughs (hur hur, Parker...do you park cars for a living? hurhur.) But both have different readings, kouenjin and chuushajin respectively. Really, they are both for laughs...
What kanji would you choose for your name? If you are interested, let me know in the comments and I will help you pick some out.