In मार्कण्डेय ‌समस्या पर्व within वनपर्व of महाभारत, there is a famous episode called popularly as the व्याधगीता. The following is a small extract of this episode which in simple terms defines the qualifying criteria for who may be called as a ब्राह्मण

य: क्रोधमोहौ त्यजति तं देवा ब्राह्मणं विदुः। यो वदेदिह सत्यानि गुरुं संतोषयेत् च।। हिंसितश्च न हिंसेत् तं देवा ब्राह्मणं विदुः।

The देव know him to be a ब्राह्मण who foesakes anger and cravings; and also speaks truth here and keeps his गुरु in contentment and who himself facing injury never injures others.

जितेन्द्रियो धर्मपरः स्वा़ध्यायनिरतः शुचिः। कामक्रोधौ वशौ यस्य तं‌ देवा ब्राह्मणं विदुः।।

The देव know him to be a ब्राह्मण who again possess control over his senses, focussed on धर्म, ever devoted to study of the वेद,  keeps himself clean and who has full control over his desires and anger.

यस्य च‌ आत्मसमो लोको धर्मज्ञस्य मनस्विनः। सर्वधर्मेषु च‌ रतस्तं देवा ब्राह्मणं विदुः।।

The देव know him to be a ब्राह्मण who aquainted with धर्म and possessed of intellect (or vigour) keeps himself balanced and who is conversant with all systems of धर्म. 

योsध्यापयेदधीयीत यजेद् वा याजयीत वा। दद्याद् वापि यथाशक्ति तं‌ देवा ब्राह्मणं विदुः।।

The देव know him to be a ब्राह्मण who himself studies while reaching others and who again performs यज्ञ himself while presiding over यज्ञ performed by others and who does charity in alignment with his resources.

ब्रह्मचारी वदान्यो योsधीयीत् द्विजपुङ्गवः। स्वाध्यायवानमत्तो वै तं‌ देवा ब्राह्मणं विदुः।।

The देव know him to be a ब्राह्मण who is a ब्रह्मचारी (detached), possessed of a charitable disposition, always attends to this studies and who carefully studies the वेद. 

यद् ब्राह्मणानाम् कुशलं तदेषां परिकीर्तयेत्। सत्यं तथा व्याहरतां नानृते रमते मनः।।

Whatever is favourable to any ब्राह्मण appeals to such a person too. The mind of those who always walk the path of truth and never takes pleasure in untruth. 

धर्मं तु ब्राह्मणस्याहुः स्वाध्यायम् दममार्जवम्। इन्द्रियाणां निग्रहं ‌च शाश्वतं द्विजसत्तमः।।

The धर्म of a ब्राह्मण is said to be स्वाध्याय (of the वेद),  control of external orientation, simplicity of conduct and ever watchful of conduct of the sense organs.

सत्यार्जवे धर्ममाहुः परं धर्मविदो जनाः। दुर्ज्ञेयः शाश्वतो धर्मः स च सत्ये प्रतिष्ठितः। श्रुतिप्रमाणो धर्मः स्यादिति वृद्धानुशासनम्।।

The people who are well versed in धर्म consider the control of senses, establishment in सत्यं‌ and simplicity of behaviour to be the eternal and highest धर्म. धर्म is eternal and difficult to attain. It is established upon सत्यं. धर्म again rests entirely on श्रुति (viz the वेद), which is the saying of old (meaning wise) men. 

Source:

Translation of Mahabharata by MN Dutt