One of my friends from US

One of my Friends came to Japan in the end of August,

=>One of my friends came to Japan at the end of August.

she is from New York, to teach English in the institute in Tokyo.

=>She came from New York to teach English in an institute in Tokyo.

She will stay in Tokyo for one year. We knew via other web page before but I haven't met her so far. I'm looking forward to the day when we can meet.

=>She will stay in Tokyo for one year. We knew each other online/through (a) website but we never met in person. (<=I think this is more natural!) I'm looking forward to the day when we can meet. 

By the way, since she loves to go cycling, she brought her bicycle and goes out by it when she has time and it doesn't rain.

=>By the way, since she loves to go cycling, she brought her own bicycle and goes out by it when she has time and when it's not raining.

Last weekend, she went around the center of Tokyo, visited one of the park and the Imperial Palace and wrote about it on her blog with beautiful photos. If you are interested in them, please take a look it.
=>Last weekend, she went around the center of Tokyo, visited one of the park and the Imperial Palace and wrote about it on her blog with beautiful photos. If you are interested in them, please take a look at it.

ブログ見ました。すごいですよ！

りょこさんの英語もすごかったよ！少しだけ直していたでど。。。

がんばってね。

One of my friends from the (US is short for United States, and since "United States" can be used to refer to other things than the US, such as in "The United States of Brazil", the definitive article must be used to specify that we mean the United States of America) US

One of my Friends from New York came to Japan in the end of August, she is from New York, to teach English in the institute in Tokyo. She will stay in Tokyo for one year. We knew eachother via another website before but I haven't met her before. I'm looking forward to the day when we can meet.

Also, since she loves to go cycling (Maybe say Biking here? Cycling works, but sounds a bit odd), she brought her bicycle and goes out with it when she has time and it doesn't rain. Last weekend, she went around the center of Tokyo, visited one of the parks and the Imperial Palace and wrote about it on her blog with beautiful photos. If you are interested in them, please take a look it.

Please correct my English and I would like to know the most natural way to phrase things in English.
Thank you.

Good job, you have really well formed sentences.

One of my friends from the U.S.

One of my friends came to Japan at the end of August. She is from New York to teach English at a  the institute in Tokyo. She will stay in Tokyo for one year. We knew each other through a webpage/through online before, but we have not met each other. I'm looking forward to the day when we can meet.

By the way, since she loves to go cycling, she brought her bicycle and goes out by it when she has time and it doesn't rain. Last weekend, she went around the center of Tokyo, visited one of the park, and the Imperial Palace, and made blog about it/blogged about it with her beautiful photos included. If you are interested in them, please take a look it!

http://t.co/tcr1oEudKO

Please correct my English and I would like to know the more natural way to say in English.
Thank you.

These two inserts that I made:

made a blog about it...

blogged about it...

-----It's what bloggers say in person. I am a blogger, too and I've seen "I made a blog about it. I blogged about it." 

If you write on paper then it is best to write "I made a blog about it."

If you want to say it--you can say both ways: I blogged about it. I made a blog about it.