First step (some simple logical deduction):
Now, if we focus on this highlighted 2x3 grid:
we see two B's. If we choose the red B we would go against the antiknight-rule in the 2x3 grid below the yellow one, so we now know we have to choose the blue B.
That leads to:
This red B. And the rest of the boxes highlighted in yellow will follow.
Let's clean this up:
Now, using some more logic, we can place:
This red D. And the rest of the highlighted yellow boxes will follow.
Let's clean up again:
We now have the whole grid in terms of letters. Time to look how we can translate these to numbers, 1-6, using the rules.
From the information given by the rules we can conclude that:
Column1 and row5 must be of the form, 2-3-4, 3-4-5 or 4-5-6
(note that the possibility 1-2-3 is excluded since 2 is 1 doubled) where its possbible along the column/row. We can see that the string of numbers must include a 4.
Let's compare column1 possibilities with the corresponding letter-position in row5 and see what happens:
This string won't work since it's impossible to form a 3 digit consecutive string in row 5.
Like-wise this string won't work since it's impossible to form a 3 digit consecutive string in row 5.
Now this one is possible if we choose them to be 3-4-5. The other possibilities (2-3-4 and 4-5-6) don't work.
Same as the two first possibilities we can see that this is impossible.
This means that the third possibility, C-B-E, is the only one that works.
We now replace the letters with the corresponding numbers and get the full grid: