It's difficult to know precisely what the public, or the aristocracy thought of someone during the 12th century. Very few people from that era wrote down their opinions. The only group of people who wrote down their opinions were chroniclers, and chroniclers during this era were typically biased and writing with an agenda. Therefore although we can come to a conclusion based on the opinion of chroniclers, a conclusion based solely on the writing of chroniclers may not be entirely reliable. A way we can check what chroniclers said about Richard, and to gauge the opinion of the public and the aristocracy is by inferring the opinion of people from the way they acted during important events.

The Lion in Winter is set in 1183 and Richard became King in 1189.

Richard the Lionheart was mainly known for his military ability, his first military campaign involved travelling through his mother's duchy of Aquitaine(in south-western France) as a show of force in order to pacify unrest. In 1172 he became the duke of Aquitaine, and in 1173 he joined his brothers in rebelling against his father. Richard was then sent to secure control of the city of La Rochelle which was in Aquitaine, however the inhabitants of the city refused to support him, and he was repulsed. This show that at this point Richard didn't have a fearsome reputation, and wasn't even completely in control of his own duchy.

The brothers were unable to even secure control over the French territory of their father, and by the time Henry II arrived in France, Richard and his brothers were already losing. Eventually Richard surrendered to his father in 1174, crying and begging for forgiveness, this scene would be bad for Richards' reputation among those who saw it, although the general public wouldn't have seen or heard about it.

In subsequent years Richards reputation significantly improved, due to his military success. In 1175 Henry sent Richard to subdue the other rebelling barons in Aquitaine. The chronicler Roger of Hoveden was especially impressed with Richards performance in the siege of Castillon-sur-Agen.

Richard gained even more reputation during his siege of Taillebourg in 1179, and in his conflict with the viscount of Limoges in 1182. However the fact that Richard had to embark on so many campaigns show that he wasn't well liked by his subjects, and that his reputation among his subjects was one based on fear.

By 1183 Richard did have a powerful military force that was capable of challenging his father, and by this point had gained a fearsome reputation as demonstrated by his success in capturing his father's capital at Chinon in 1187 without a battle. This shows that by this point Richard had a fearsome reputation.

Richard the Lionheart's reputation and intimidation was helped by him having a very impressive appearance, with both primary and secondary sources describing him as very tall. With Clifford Brewer claiming that he was 196 cm tall(6 foot 5 inches). Since Richard was mainly known for his military ability, his impressive size no doubt helped his reputation with people who met him.

In summary although early in his life Richard had some embarrassments that harmed his reputation, by 1183 Richard had a legendary reputation based on intimidation, and military prowess among both the aristocracy and the public.

Sources:

The Death of Kings: A Medical History of the Kings and Queens of England by Clifford Brewer

Richard the Lionheart: King and Knight by Jean Flori

The Reign of Richard Lionheart by Ralph Turner

Itinerarium Regis Ricardi by Richard de Templo

Gesta Regis Henrici II & Gesta Regis Ricardi by Roger of Hoveden