Two phase clock on a breadboard
Use a 2-bit binary counter (two D flip-flops), then use some simple combinational logic to generate the phase1/phase2 outputs. This could be individual AND / OR / NOT gates, or it could be a decoder such as 74__139.

There are four ways to map the counter state to the phase1/phase2 waveforms, you can choose whichever seems the easiest to implement. Remember to consider the initial state of the counter as well.

At 100kHz there shouldn't be any major issues with using solderless breadboard. Up around 10MHz solderless breadboard starts to reach its limits.


------|-------|------
a1 a0 |phase1 |phase2
------|-------|------
0  0  |0      |0
0  1  |0      |1
1  0  |1      |1
1  1  |0      |1
0  0  |0      |0
0  1  |0      |1
1  0  |1      |1
1  1  |0      |1



phase1 = a1 AND (NOT a0)
phase2 = a1 OR a0  


------|-------|------
a1 a0 |phase1 |phase2
------|-------|------
0  1  |0      |0
1  0  |0      |1
1  1  |1      |1
0  0  |0      |1
0  1  |0      |0
1  0  |0      |1
1  1  |1      |1
0  0  |0      |1



phase1 = a1 AND a0
phase2 = a1 OR (NOT a0)  


------|-------|------
a1 a0 |phase1 |phase2
------|-------|------
1  0  |0      |0
1  1  |0      |1
0  0  |1      |1
0  1  |0      |1
1  0  |0      |0
1  1  |0      |1
0  0  |1      |1
0  1  |0      |1



phase1 = a1 NOR a0
phase2 = a1 AND (NOT a0)  


------|-------|------
a1 a0 |phase1 |phase2
------|-------|------
1  1  |0      |0
0  0  |0      |1
0  1  |1      |1
1  0  |0      |1
1  1  |0      |0
0  0  |0      |1
0  1  |1      |1
1  0  |0      |1



phase1 = (NOT a1) AND a0
phase2 = a1 NAND a0