Just from having a quick look at imgur and this question, I've come up with (pretty much just combined the two) the following. Let me know if it doesn't work.
Bitmap bitmap = yourBitmapHere;
// Creates Byte Array from picture
ByteArrayOutputStream baos = new ByteArrayOutputStream();
bitmap.compress(Bitmap.CompressFormat.PNG, 100, baos); // Not sure whether this should be jpeg or png, try both and see which works best
URL url = new URL("http://api.imgur.com/2/upload");
//encodes picture with Base64 and inserts api key
String data = URLEncoder.encode("image", "UTF-8") + "=" + URLEncoder.encode(Base64.encode(baos.toByteArray(), Base64.DEFAULT).toString(), "UTF-8");
data += "&" + URLEncoder.encode("key", "UTF-8") + "=" + URLEncoder.encode(YOUR_API_KEY, "UTF-8");
// opens connection and sends data
URLConnection conn = url.openConnection();
conn.setDoOutput(true);
OutputStreamWriter wr = new OutputStreamWriter(conn.getOutputStream());
wr.write(data);
wr.flush();
Edit: It seems we need to pass Base64.DEFAULT as the second option to Base64.encode. Updated the example above.
Edit 2: Can you use the following code, based upon the oracle site, and report back what it outputs:
BufferedReader in = new BufferedReader(
new InputStreamReader(
conn.getInputStream()));
String inputLine;
while ((inputLine = ic.readLine()) != null)
System.out.println(inputLine);
in.close();