Hill's team, who were pre-season relegation favourites with several bookmakers, are two wins away from being crowned Premiership champions.
Their progress into the title play-offs means they have already surpassed last summer's twin targets of 50 points and a finish somewhere between fifth and eighth.
Bristol finished third after 22 games - only table-toppers Gloucester won more Premiership matches this term - but a huge challenge now awaits in the form of treble-chasing Leicester.
The Tigers are building ominously towards achieving an unprecedented trophy hat-trick, having already lifted the EDF Energy Cup, while victory over Bristol would take them into next week's Premiership final and they are already through to a May 20 Heineken Cup final appointment with Wasps.
Bristol will find themselves in a familiar positon of underdogs when they arrive at Welford Road, yet head coach Hill retains a strong sense of optimism.
He said: "We have a great opportunity to get to a marvellous occasion - a final at Twickenham.
"By the time the game comes, our players will believe 100% they can win at Welford Road.
"Everyone knows that strange things can happen to teams in cup games. The unexpected has been known to happen, and people can win matches against the odds."
Bristol though, will be without influential back-row forward Andrew Blowers because of a knee ligament injury, so Nathan Budgett deputises, while a calf muscle injury sidelines centre Brian Lima, whose place goes to Nathan Brew.
Leicester make three changes after demolishing Wasps last weekend, with head coach Pat Howard calling up wing Tom Varndell, prop Alex Moreno and lock Leo Cullen.
Howard said: "They beat us a few weeks ago. They are a good side and they are going to be very hard to beat.
"It is going to be fascinating - we have the league's best attack, and they have the league's best defence."