How do I use the character 当, pronounced as a 4th tone?
According to《规范》one of the meanings of 


  当 
  
  dàng


is


  以为


and they give the sample sentence:


  我当你不来了呢


So it seems the difference between 当 and 以为 is minimal at best.



As for other example sentences, here's a gloss of the PLC dictionary provided by Pleco:


  ADJECTIVE
  
  1 proper; right; appropriate
  
  以上意见当否, 请批示。 
  Yǐshàng yìjian dàng fǒu, qǐng pīshì.
  We are awaiting your comment on the proposals set forth above. 
  
  2 that very (day, etc.)
  
  当李的产量 
  dàng lǐ de chǎnliàng
  the output of that season 
  
  3 of the same; of one's own
  
  当村的小伙伴们 
  dàng cūn de xiǎo huǒbàn men
  little friends of one's village 
  
  VERB
  
  1 match; be equal to
  
  他一个人能当两个人用。 
  Tā yī ge rén néng dàng liǎng ge rén yòng.
  He can do the work of two persons put together. 
  
  2 treat as; regard as; take for
  
  不要把支流当主流。 
  Bùyào bǎ zhīliú dàng zhǔliú.
  Don’t take minor aspects for major ones. 
  
  3 think
  
  我当你不知道呢。 
  wǒ dàng nǐ bùzhīdào ne.
  Well, I thought you didn’t know. 
  
  4 pawn
  
  当衣服 
  Dàng yīfu
  pawn one’s clothes; put one’s clothes in pawn 


Hopefully that gives you a better idea of how it's used.