I hope that you will consider my application. I'm hoping that you will (you'll) consider my application. Either or. Though, 'would' is something that has either happened already or has yet to happen. Will is more in the 'now.' At least that's how I understand it. Maybe someone else can come along and correct me/explain further.

"I hope / I am hoping that you would consider my application." Both are fine; however, "I hope that you would consider my application." sounds a bit more native - good pragmatics too.

Both Dayn and Nick are right