If you are already dating someone, your situation is different. But you should not be converting to Judaism for that person. You need to think about what will happen if you break up: will you still want to be Jewish? Will you find another Jewish partner and still raise your children Jewishly? Conversion is permanent, are you OK with that regardless of this relationship status? Converting while dating can also be problematic because the converting partner tends to become more observant and/or interested in Judaism. But that has its positive side as well.
Starting a new relationship while converting is not a good idea. When you're converting, you are building a relationship with Judaism, with your rabbi, the members of your shul, the Jewish community, with Israel and maybe reestablishing or reevaluating a lot of your other relationships. And, of course, you are working on your relationship with G-d. You have A LOT going on. Adding another new relationship into that can be messy.
You might not be as dedicated to the new relationship as you would be otherwise. You might not be alert to signals about where the relationship is or isn't going. If your partner isn't Jewish, he or she might not understand that you will eventually be Jewish and your relationship will be an interfaith one. If your partner is Jewish, he or she might have a different observance level than you and might be resistant to your way of doing things. Either way, your whole relationship with this partner is intertwined with your conversion to Judaism. If and when the relationship ends, you'll realize a lot of your really great Jewish memories are associated with them. And you might want to forget those things, but you can't because they are some of your earliest and most important Jewish memories.
If this person really is "the one," they'll be there after you convert. It only takes about a year to convert to Judaism and you have the rest of your life to find your Jewish soul mate and build a Jewish life with them. Don't rush things because you might end up really, really hurt.