Why did Lord Krishna attack Bhishma (Bhishma Pitamaha), in spite of vowing that he would not pick up a weapon in the Kurukshetra war?
Shri Krishna had to attack Bhishma (actually He almost attacked him) because Arjuna was unable to stop the wrath that Bhishma had unleashed. Arjuna promised Shri Krishna to take out Bhisma in battle:
चॊदयाश्वान यतॊ भीष्मः करिष्ये वचनं तव
पातयिष्यामि दुर्धर्षं वृद्धं कुरुपितामहम [MB - 6.102.37]
Urge the steeds, O Hrishikesa, to where Bhishma is. I will do as you say. I will overthrow the Kuru grandsire Bhishma, that invincible warrior.
But while battling with Bhisma, Arjuna broke the string of his own bow:
अथास्य तद अपि करुद्धश चिच्छेद धनुर अर्जुनः 
So Shri Krishna seeing that Arjuna was fighting mildly and Bhishma was showering arrows, put away the reins of the horses and jumped down from the chariot and proceeded to kill Bhishma:
वासुदेवस तु संप्रेक्ष्य पार्थस्य मृदुयुद्धताम
भीष्मं च शरवर्षाणि सृजन्तम अनिशं युधि 
उत्सृज्य रजतप्रख्यान हयान पार्थस्य मारिष
करुद्धॊ नाम महायॊगी परचस्कन्द महारथात
अभिदुद्राव भीष्मं स भुजप्रहरणॊ बली
परतॊदपाणिस तेजस्वी सिंहवद विनदन मुहुः
दारयन्न इव पद्भ्यां स जगतीं जगतीश्वरः
करॊधताम्रेक्षणः कृष्णॊ जिघांसुर अमितद्युतिः
गरसन्न इव च चेतांसि तावकानां महाहवे [MB 6.102.53-55]
Abandoning then, O sire, Partha's silver like steeds, and filled with wrath, that great lord of Yoga powers jumped down from that great chariot. Repeatedly roaring like a lion, the mighty Krishna of great energy and immeasurable splendor, the Lord of Universe, with eyes red as copper from rage, and having his bare arms alone for his weapons, rushed towards Bhishma, whip in hand, desirous of slaying him and seeming to split the universe itself with his tread.
But after that, Arjuna pleaded Krishna to not break His vow and promised to kill Bhisma himself. Shri Krishna in anger without saying a word to Arjuna (unlike what we have seen in TV shows) goes back and mounts the chariot.
माधवस तु वचः शरुत्वा फल्गुनस्य महात्मनः
न किं चिद उक्त्वा सक्रॊध आरुरॊह रथं पुनः 
Madhava, however, hearing these words of the high-souled Phalguni, spoke not a word, but in anger once more mounted upon the chariot.
So from this account of the Mahabharat, it seems that even though Shri Krishna had promised to not use weapons, he was forced to take the step of using one due to the unstoppable wrath of Bhishma which Arjuna had failed to prevent.
On a side note, Shri Krishna did this because His main purpose was not to be known as someone who keeps his vow, but to establish dharma as He had stated to Arjuna in Gita (4.7-8). So when He saw the side of dharma was facing trouble in the battle, He had to interfere in it anyway. Also, He might have done it to please Bhishma as well who was His devotee.
Reference: Mahabharat - Bhishma Parva