While reciting Talbiyah, enter Masjid al-Haram with right foot first. Supplicate to Allah and proceed towards Ka' bah. You may recite the following supplication which is usually used before entering a mosque:
اللَّهُمَّ افْتَحْ لِي أَبْوَابَ رَحْمَتِكَ
" Allah huma aftah li abwabe rahmate ka"
(Oh Allah open for me the doors of your mercy.)
At the first sight of Ka' bah, keep your eyes fixed at Baitullah and standing at one side, recite the following dua:
اللّهُ أكبر اللّهُ أكبر, لا إلَهَ الا اللّه, وَاللّهُ أكبر اللّهُ اكبر, و لِلّه الحمدَ
" Allahu Akbar, Allahu Akbar, La Ilaha Illallahu, Wallahu Akbar, Allahu Akbar, Walillaahil Hamd"
(Allah is the greatest, Allah is the greatest, Allah is the greatest, there is none worthy of worship except Allah and Allah is the greatest and to Allah is all praise.)
Proclaim Darud on our beloved Prophet (peace be upon him) and very humbly and with tears in your eyes supplicate to Allah for whatever you wish. This is a special time for the acceptance of prayers.
The idea is to praise and glorify your Creator before proclaiming Darud and supplications. Therefore, in lieu of Allahu Akbar and La Ilaha Illallah, you may recite some other similar holy verses if you so desire.
After this, while uttering Talbiyah, move forward to perform Tawaf of Ka' bah.