If you just want a hint, read a question and not the answer.
Question: What number can go in C5?
C5 is in the same column as 15, and is less than 15 (Clue A1). C5 must also be greater than B4 (Clue D5). Since 14 is the only number between 13 and 15, C5 = 14.
Question: What numbers go in the cells on or above the A1-E5 diagonal?
There are 15 total numbers on or above the A1-E5 diagonal. Each number in column 5 must be 15 or less, and each number in a diagonal above the A1-E5 diagonal must be less than a number in column 5 (Clue D5). Thus all numbers on or above the A1-E5 diagonal must be 15 or less. Since there are 15 of these numbers, 1-15 appear on or above the diagonal, and 16-25 appear below the diagonal.
Question: What number can go in C1?
C1 is a multiple of 6 and is greater than 15. Thus it is either 18 or 24. We know that both D2 and E3 must be greater than C1 (Clue D5). Since two numbers in the puzzle are greater than C1, C1 cannot be 24, and thus must be 18.
Question: What number can go in B1?
B1 is below the main diagonal, and is thus between 16 and 25. B1 is also less than C1 (Clue C5). Thus B1 is either 16 or 17. Since every number in the B row is prime, B1 must be prime, and thus must be 17.
Question: What are the possibilities for the remaining 3 numbers in the B row?
The four remaining prime numbers less than 15 are 2,3,5,11. Exactly 3 of them appear in the B row. 2 cannot be in B3 or B5, because both of these cells are greater than a cell in the A row, and 1 has already been placed. If 2 is in the B row, it must be in B2. In this case, 3 cannot also appear in the B row, because 3 would need to be greater than a cell in the A row, and 1 and 2 would not be available. Thus we conclude that 2 and 3 cannot simultaneously be in the B row. Since the prime number missing from the B row is either 2 or 3, both 5 and 11 MUST appear in the B row.
Question: Where can 11 go?
We know 11 appears in the B row. 11 cannot be placed in B2, because A3, B4, and E5, must all be greater than B2 and still be less than 15. The only unplaced number greater than 11 and less than 15 is 12. If 11 were in B3, C4 would need to be 12, but this violates Clue B4. Thus 11 cannot be in B3 and must be in B5.
Question: What number can go in B3?
B3 is a prime less than 15, so it must be either 2,3 or 5. B3 cannot be 2 because it needs to be greater than A2, and 1 has already been placed. B3 cannot be 3 because of the clue in B5. Thus B3 is 5.
Question: Where can 12 go?
There are no remaining numbers between 12 and 15 to be placed. Thus 12 cannot be placed in C3 or D4, because E5 needs to be greater than C3 and D4, but less than 15. Additionally 12 cannot be placed in C4 (Clue B4) or in the A row (Clue B3), so 12 must be placed in E5.
Question: Where can 6 go?
6 cannot go in the A row because it is a multiple of 3. 6 cannot go in B2 because it is not prime. 6 cannot go in C3 because it is even. 6 cannot go in C4 because it is a multiple of 6. Thus 6 goes in D4.
The only odd number left between 1 and 6 is 3, so 3 goes in C3. The only prime left is 2, so 2 goes in B2. The only number less than 5 left is 4, so 4 goes in A2. 9 is a multiple of 3, so it cannot go in the A row, so it must go in C4.
Question: Where can 16 go?
16 is smaller than 17 and 18, so it cannot share a diagonal with either of them. Thus 16 is either in D1, E1, or E2. 16 cannot go in D1 because it is even (Clue C4). 16 is also the smallest number greater than 15, so if it went in E2, there would not be a smaller number for D1. Thus 16 cannot go in E2 and must go in E1.
Question: Where can the rest of the multiples of 4 go?
8 cannot go in A3 because it would be adjacent to 4, so 8 is in A4 and 10 is in A3. The remaining multiples of 4 are 20 and 24. The remaining spaces not adjacent to any multiples of 4 are C2, D3, and E3. If a multiple of 4 went in D3, there would be no place for the other multiple of 4 to go, so D3 cannot contain a multiple of 4. Thus 20 and 24 go in C2 and E3. Since both D3 and E4 must be greater than C2, 24 cannot go in C2. Thus 24 goes in E3, and 20 goes in C2.
The only unplaced even number is 22, so 22 goes in E4. The only number between 20 and 22 is 21, so 21 goes in D3. Since 25 is the biggest number, it cannot go in D1 or D2, and must go in E2. The only remaining unplaced number greater than 20 is 23, so 23 goes in D1. 19 is the last remaining number, so it goes in D2.
You made it! Well done.