Float number to binary
If I understand your question correctly, you want to determine the binary representation of a floating-point number by hand. I assume you are talking about the IEEE 754 floating-point format, since this is the most common format.

You can indeed do this by successively multiplying with 2. However, just multiplying does not result in anything useful, you should also subtract one (1) every time the intermediate result is greater than or equal to 1. To understand this, you must understand how floating-point numbers are stored. This is explained pretty well on Wikipedia (and probably many other sources), including the algorithm you mean (see here).

To do this by hand is quite painful indeed. It's normal that you have to do more than 20 multiplications. You first need a few multiplications to normalize the number and determine the exponent (i.e. scale the number such that it has the form 1.xxxxx * 2^exponent). After that, each multiplication yields exactly one bit of the mantissa. A single-precision floating-point number has a 23-bit mantissa, so to convert the number 0.347 to single-precision floating-point, you need to do 25 multiplications. A double-precision floating-point has a 52-bit mantissa, so you'll need 54 multiplications!

NB: For sake of simplicity, I did not take into account numbers larger than 1, negative numbers, special numbers (such as NaN, +/- infinity, denormal numbers, etc.) and other aspects such as rounding. These cases seem to be out of the scope of your question.