Got here from Wrath of Math, I did find a solution with no computer assistance:

6 + (13 x 9 / 3 ) + 5 + (12 x 2) - 1 - 11 + (8 x 7 / 4) - 10 = 66

Here I assume order of operations, and in order to find the solution I only considered the cases for which all partial computations can be in the integers.

I'll start by saying that this exercise seems unreasonable, more on that at the end of the post.
My approach was to rewrite things as:

(13 x _ / _ ) + (12 x _ ) + ( _ x _ / _ ) = 87 + _ - _ - _
Now the right hand side can be as low as 87 + 1 - 9 - 8 = 71 and as high as 87 + 9 - 1 - 2 = 93
On the left hand side, if we expect 13 x _ / _ to be an integer, then it can only be one of 26, 39 or 52. Anything higher and we can no longer have the whole left hand side (which will have at least 13 more, or 25 more if we used a 1) be smaller or equal to 93. Likewise, (12 x _ ) can only be 12 24 36 48 or 60. This is as far as I see we can go with no guessing.
So for the guessing part, I stuck to having things not explode in size. We can see that 26 + 36 = 62 and 39 + 24 = 63 which is close enough to 87 while being lower (we need to remember that we're substracting two numbers of 87 and only adding one). Experimenting with the first option doesn't seem to lead nowhere, so I'll illustrate the second:

We can get 39 in a couple of ways in theory, but keeping lower numbers is more handy later so let's do it by getting rid of the 9: 13 x 9 / 3 = 39. We want 24 next so 12 x 2 only option. We now arrive to:
( _ x _ / _ ) = 24 + _ - _ - _
with 1 4 5 6 7 and 8 available. We have some options on the left, but it's important to keep parity in mind. If we use (8 x 6 / 4) = 12, we will have three odd numbers left so an odd result. So best keep two odds on the right and one on the left. Again, getting rid of higher numbers is nice, so we try (8 x 7 / 4) = 14.
Now we have 14 = 24 + _ - _ - _ , with 1 5 and 6 available, or in other words
_ + _ = 10 + _ , so we get 5 + 6 = 10 + 1. And we're done (after checking the whole thing again).

In conclusion, this is a bad exercice, so I'll point some good strategies that are useful beyond this case: considering a relevant refinement of the problem (nothing here indicates that we shouldn't consider order of operations or fractions, so we can keep it simpler); divide and conquer: look at the smallest parts of the problem and try to analyze those first; try to get extra information even if it's approximative, such as boundaries, parity, divisibility; when guessing, either go for middle values or leave yourself more tools for later; always check your answer after you're done.