All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
You are not permitted to conduct the marriage contract following anyway other than the way legislated by Allah. You are a Muslim and a Muslim is obliged to conduct the marriage contract according to the Sharee'ah. Non-Islamic authorities do not implement the conditions and basic elements set by the Sharee'ah. Therefore, conducting a marriage contract without fulfilling the rules of the Sharee'ah makes it void.
Know that a marriage is considered valid in the Sharee'ah when it meets the basic conditions which are as follows:
1) Guardian's approval.
2) Witnesses (2).
3) Mahr (bridal money).
4) It should be free from the legal prohibitions, such as the woman being married, or she is in ‘Iddah (waiting period after divorce or death of her husband).
Since her family does not approve your marriage, you are not permitted to marry her.
Therefore, it is not permissible for you to have intimate relations with her unless you marry her again fulfilling the conditions set by the Sharee'ah.
In addition, you should consult an Islamic Center near you. The authorities there will explain the matter in detail.
Allah knows best.