Hi, I'm making a monk character and want a interesting background. I'm thinking a urchin who is picked up by a traveling monk sort of thing but open to ideas. 

Base it on the story you built for that character.

My monk left is noble family to become a woodworker. Later, his adoptive family was killed and he sought justice and became am urban bounty hunter. I had 2 backgrounds to choose from, but decided on the bounty hunter as that was a deciding moment in his life.

He left the noble background and chose the bounty hunter life. So, mechanics aside, what defines your monk?

My halfling monk also has the urchin background.  His history is that he was a inner city halfling in a large family that was very poor.  He turned to burglary/petty theft to originally help his family but fell into the wrong crowd and got caught by the local sheriff on one to many times.  The Law gave him a choice to be imprisoned for an untold amount of years, which may cause more grief for his family, or join the local monastery as part of a community service to work off his crimes and learn a more productive life style.  His skills at hiding/sneaking were higher than normal and gained the attention of the specialized sect Shadow monks within the monastery.  From there he learned to pretty much be a ninja (darkness/invisibility/teleport) and I am having the very good time playing him.

That's great! I was thinking along the same lines.

The backgrounds are of all parts the most flexible for players.  Pick things that are thematically in line with your plans and go team go, don't agonize too long over the background in particular.

yeah

My monk is an urchin who was brought up on the streets, running scams, pick pocketing and so on.
He was trained by a monk that stayed nearby in the poor end of the city. 

Haunted one...

Dhampir who left many victims in her wake. She can still hear their voices and feel their presence. Eventually she sought out a monastery that taught her to use her ki power to control her blood urges and push away the voices... mostly. 

Hi. 

I used criminal background as my monk was in the traveling circus. He was taught a few tricks like slight of hand to keep people entertained and loose out on money. He also met a two headed monk, who eventually helped set this hero on a path of perfection for mind and body. 

Although his main passion in the circus was fighting (or acrobatics or what ever), he was used primarily as laborer. So he knows his way with carpentry tools (or other tools). When he did fight, he took the falls and hits to help other fighters look good. This has since cause some memory loss (role playing low intelligence) and his eventual firing. 

This character now roams as an adventurer to establish his name as a fighter. What ever he finds can help finance his goals and the people will know his heroics (Bond). 

I have greatly enjoyed this character as it can be played any way you want. By third level you can decide if she has perfected fighting skills with open hand, heal the brain through ki with way of mercy, or dive into a drunken master. With criminal used as background, any tool set can be chosen. Doesn't have to be thief's tools or carpentry because anything goes at a circus. This allows the character to be replayable (in my mind), and easy to multi class for what ever reasons. 

Like improving performance as a bard or stealth as a rouge. 

When I was trying out 1D&D I used the Guide background on a monk, and I really enjoyed it.
Played him as one of those wandering ascetic monks, who would lead travelers through mountain passes for a few coins, and had just three spells from the background (Shillelagh worked well with a walking stick, and I used Goodberry to imitate a fasting).

mine is a tabaxi spy who acts as the butler for a drug cartel. the cartel heads 8 year old daurghter has been kiddnapped. to stop the parents from starting a blood bath he goes to an island to find her and things get more complicated when her soul is split from her body. 

Mine is a Way of Mercy half-elf who was raised by rats underneath the main city. Through a calling from a higher power, he learned everything he knew whilst meditating.

I made my monk an arena combatant who was also descended from escaped gladiators. He wins a local tournament and attains a dragon mark. he then goes on a search looking for his rival.

He is a panther like tabaxi Kensei, and his rival is a swift stride shifter themed to look like a white puma.