どうか details

write + (and…) + please, I beg you — a request made from a position of having no claim on the listener

douka kaite kudasai

please, I beg you — a request made from a position of having no claim on the listener

meaning: please do write

事情じじょうは後あとで必かならず説明せつめいしますから、どうか信じしんじてください。

jijou wa ato de kanarazu setsumei shimasu kara, douka shinjite kudasai.

I will explain everything later, so please do believe me. neutral

明日あしたの試験しけん、どうかうまくいきますように。

ashita no shiken, douka umaku ikimasu you ni.

Let tomorrow's exam go well, please. neutral

Examples

please, I beg you — the speaker asks for something they cannot demand

事情じじょうは後あとで必かならず説明せつめいしますから、どうか信しんじてください。

jijou wa ato de kanarazu setsumei shimasu kara, doukashinjitekudasai.

I will explain everything later, so please do believe me. neutral

明日あしたの試験しけん、どうかうまくいきますように。

ashita no shiken, douka umaku ikimasu you ni.

Let tomorrow's exam go well, please. neutral

寒さむい日ひが続つづきますが、どうかお体からだを大切たいせつになさってください。

samui hi ga tsuzukimasu ga, douka o karada o taisetsu ni nasatte kudasai.

Cold days continue — please do look after yourself. formal

遠とおくにいても、どうか元気げんきでいてください。

tooku ni ite mo, douka genki de ite kudasai.

Even though you are far away, please do take care of yourself. neutral

どうかしている is a fixed idiom of its own — 「君きみ、どうかしてるよ」 says
something is wrong with someone, and carries none of the pleading here.

なんとか
どうか asks for goodwill the speaker has no claim on; なんとか asks for a way — どうか begs for the act, なんとか begs for the means that would make the act possible.
ぜひ
Both press a request, and they differ in whose interest is at stake — ぜひ says the act would be good for the listener and belongs to invitations, どうか says the speaker is at the listener's mercy and belongs to apologies and pleas.

かどうか
The same three kana cut a different way — かどうか belongs to the clause in front of it and asks whether something holds (行いくかどうか), while どうか leans forward onto the request that follows and asks for nothing to be decided.

てください
ください on its own carries instructions and shop signs; どうか in front puts the speaker's whole weight behind the request, so the pair belongs to apologies and appeals and sounds overwrought on an instruction.