You should think about what you want to acheive. If you terminate the father's parental rights, he will never be required to pay child support. You can ask him to voluntarily give up his rights or you can sue him and ask the court to terminate. Going through the court can be expensive and you have show legal reasons why he is not entitled to keep his parental rights. Are married or divorced from the father?
You really need an attorney to help you sort this out.
I practice law in Harris County, Texas so I'm most familiar with those courts.
However, 6 months is not long enough to usually terminate parental rights.
I would contact an attorney in your county that does primarily family law (aka divorce law) and pay for an hour of their time for a consultation.
You really need a face-to-face meeting in order to understand all of the options available to you.
As the other attorney stated, terminating a person's rights takes a long time and usually costs a lot of money. Why? Because once a person's rights are terminated they are considered "dead" and are no longer a parent of the child. Since this action is so severe & because of numerous appeals, the courts make sure that they do everything right. Otherwise, the higher court could reverse the decision. Most judges do NOT like to have a superior court reverse their opinions.
To answer the question of "how long" - it can take as little as a few months or as long as a few years to terminate a father's rights involuntarily. If he agrees, some courts will allow a termination without a step-father to take his place. Some courts will not allow this and will only do a termination in conjunction with a step-parent adoption - it depends on the court and the judge, although both are technically available to you. In Hardin County, neither Judge likes an absent father - so your best bet is to file a motion to enforce for child support and try to kick this guy's rear a bit so that he actually tries to be involved in the child's life.